Here is a list of all codes that belong to the classical domain.

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  1. 120-cell code[1]
    Spherical \((4,600,(7-3\sqrt{5})/4)\) code whose codewords are the vertices of the 120-cell. See [3][2; Table 1][4; Table 3] for realizations of the 600 codewords.
  2. 24-cell code[1]
    Spherical \((4,24,1)\) code whose codewords are the vertices of the 24-cell. Codewords form the minimal lattice-shell code of the \(D_4\) lattice.
  3. 600-cell code[1]
    Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [5; Table 1][4; Table 3] for realizations of the 120 codewords. A realization in terms of quaternion coordinates yields the 120 elements of the binary icosahedral group \(2I\) [6].
  4. Accumulate-repeat-accumulate (ARA) code[7]
    A generalization of the RA code in which the outer repetition-code encoding step is augmented with an acumulator acting on a fraction of the incoming bits. In addition, the code may be punctured after the final acumulating step.
  5. Additive \(q\)-ary code
    A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword.
  6. Alamouti code[8]
    The simplest OSTBC, with \(n=2\) transmitting antennas, \(m=1\) receiving antennas, and \(t=2\) uses.
  7. Algebraic LDPC code
    LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [9,10] or an algebraic structure such as a combinatorial design [11–13], balanced incomplete block design [14], a partial geometry [15], or a generalized polygon [16,17]. The extra structure and/or symmetry [18] of these codes can often be used to gain a better understanding of their properties.
  8. Algebraic-geometry (AG) code[19–21]
    Binary or \(q\)-ary code or subcode constructed from an algebraic curve of some genus over a finite field via the evaluation construction, the residue construction, or more general constructions that yield nonlinear codes. Linear AG codes from the first two constructions are also called geometric Goppa codes.
  9. Alternant code[22–25]
    Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant code is the \(GF(q)\)-subfield subcode of the dual of \(C\); see [26; Ch. 12]. Its parity-check matrix is an alternant matrix.
  10. Analog code a.k.a. Code over \(\mathbb{R}\).
    Encodes states (codewords) into coordinates in the \(n\)-dimensional (real or complex) coordinate space (\(\mathbb{R}^n\) or \(\mathbb{C}^n\)). Such codes include sphere packings, tilings, and any other codes that use real or complex numbers for encoding. The number of codewords may be infinite because the coordinate space is infinite, so various restricted versions have to be constructed in practice.
  11. Annealing-based spherical code[27–29]
    Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure.
  12. Anticode[30,31]
    Code for which the distance between any two codewords is less than or equal to some value \(\delta\) called the maximum distance. Anticodes can be used to construct codes that saturate the Griesmer bound; see Refs. [26,32,33] for more details.
  13. Antipode lattice[34]
    Lattice constructed via the antipode construction.
  14. Array code a.k.a. RAID code, Disk array code.
    Matrix code over \(GF(q)\) designed for use in distributed storage, with the first such application being a RAID-type array of hard-drives such that information is protected against erasure of one or more hard drives. Each column of a matrix codeword is typically treated as a single codeword coordinate via subpacketization (defined below) and represents a large data block. Array codes are denoted by \((n,k,m)\), where \(n\) is the number of nodes, \(k\) is the number of nodes needed to recover the data, and \(m\) is the column dimension of each codeword a.k.a. the subpacketization level.
  15. Array-based LDPC (AB-LDPC) code[35,36]
    QC-LDPC code constructed deterministically from a disk array code known as a B-code. Its parity-check matrix admits a compact representation [37] and is related to RS codes.
  16. Availability code[38,39]
    A \(t\)-availability parallel-recovery code is a code such any \(t\) coordinates can be recovered in multiple ways. That way, the code accomodates nodes that may be inaccessible during the recovery procedure.
  17. B-code[40]
    The first array code, constructed over \(GF(q)\). See [41] for more details.
  18. Balanced code[42]
    An even-length-\(n\) \(q\)-ary code whose nonzero codewords all have a Hamming weight of \(n/2\). A code is \(\epsilon\)-balanced if the relative weight (i.e., weight divided by \(n\)) of all nonzero codewords lies in the interval \([\frac{1-\epsilon}{2},\frac{1+\epsilon}{2}]\). A code is \(\gamma\)-unbiased if the relative weight lies in the interval \((\frac{1}{2}-\frac{1}{n^{\gamma}},\frac{1}{2}+\frac{1}{n^{\gamma}})\).
  19. Barnes-Wall (BW) lattice[43,44]
    Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known. Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively.
  20. Batch code[45]
    Binary code designed for minimizing the total amount of storage and the worst-case maximal load on any devices in a distributed system.
  21. Ben-Sasson-Goldreich-Harsha-Sudan-Vadhan (BGHSV) code[46]
    Locally testable \([n,k,d]\) code with \(n = k^{1+\epsilon}\) and query complexity of order \(O(1/\epsilon)\) for any \(\epsilon > 0\).
  22. Ben-Sasson-Sudan code[47]
    Locally testable \([n,k/2,d]_{2^m}\) code with \(k\) a power of two, \(n = k \log^{c} k\), and query complexity \(\log^{c}k\) for some universal constant \(c\).
  23. Ben-Sasson-Sudan-Vadhan-Wigderson (BSVW) code[48]
    Locally testable \([n,k,d]\) code with \(n = k \cdot 2^{\tilde{O}(\sqrt{\log k})}\) and asymptotically constant query complexity, where \(\tilde{O}(f)=O(f\cdot (\log f)^c)\) for some fixed constant \(c\).
  24. Berlekamp code[49; Ch. 9]
    A linear \(p\)-ary code that has Lee distance 5 and whose construction resembles that of RS codes. It is obtained by first constructing an RS-like parity-check matrix out of a certain field extension of \(GF(p)\) and then taking the subfield subcode of the corresponding code; see [50; Ch. 10.6].
  25. Best \((10,40,4)\) code[51,52]
    Binary nonlinear \((10,40,4)\) code that is unique [53]. Under Construction A, this code yields \(P_{10c}\), a non-lattice sphere packing that is the densest known in 10 dimensions [54][55; pg. 140].
  26. Binary BCH code[56–58]
    Cyclic binary code of odd length \(n\) whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\) (see Cyclic-to-polynomial correspondence). More precisely, the generator polynomial of a BCH code of designed distance \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=2^r-1\) for some \(r\geq 2\).
  27. Binary PSK (BPSK) code[59] a.k.a. Binary antipodal modulation, Phase-reversal keying (PRK), Antipodal signaling.
    Encodes one bit of information into a constellation of antipodal points \(\pm\alpha\) for complex \(\alpha\). These points are typically associated with two phases of an electromagnetic signal.
  28. Binary antipodal code a.k.a. Binary signal constellation.
    An \((n,K,4d/n)\) spherical code obtained from a binary \((n,K,d)\) code via the antipodal mapping.
  29. Binary balanced spherical code
    An \((n-1,K,\frac{nd}{nw-w^2})\) spherical code obtained from a constant-weight-\(w\) binary \((n,K,d)\) code via a component-wise binary balanced mapping (also known as the CW\(_2\) construction), \begin{align} \begin{split} 0&\to\sqrt{\frac{w}{n\left(n-w\right)}}\\1&\to -\sqrt{\frac{n-w}{nw}}~. \end{split} \tag*{(1)}\end{align} This construction can be extended to the general balanced binary construction CW\(_q\) for spherical code alphabets of size \(q\) [60; Sec. 6.6].
  30. Binary code
    Encodes \(K\) states (codewords) in \(n\) binary coordinates and has distance \(d\). Usually denoted as \((n,K,d)\). The distance is the minimum Hamming distance between a pair of distinct codewords.
  31. Binary duadic code[61]
    Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist for lengths \(n\) that are products of powers of primes, with each prime being \(\pm 1\) modulo \(8\) [62].
  32. Binary group-orbit code[63,64]
    Bianry legnth-\(n\) whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the group of bit-string permutations and translations, i.e., the automorphism group of binary codes under the Hamming distance.
  33. Binary linear LTC
    A binary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\).
  34. Binary quadratic-residue (QR) code
    Member of a quadruple of cyclic binary codes of prime length \(n=8m\pm 1\) for \(m\geq 1\) constructed using quadratic residues and nonresidues of \(n\).
  35. Binary-ternary mixed code[65]
    Encodes \(K\) states (codewords) in a string of \(n_1+n_2\) coordinates, with the first \(n_1\) coordinates being binary, and the last \(n_2\) coordinates being ternary.
  36. Biorthogonal spherical code a.k.a. Cross polytope code, Hyperoctahedron code, Orthoplex code, Co-cube code.
    Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope).
  37. Block LDPC (B-LDPC) code[66]
    Member of a particular class of irregular QC-LDPC codes with efficient encoders.
  38. Block code
    A code intended to encode a piece, or block, of a data stream on a block of \(n\) symbols, with each symbol taking values from a fixed alphabet \(\Sigma\).
  39. Body-centered cubic (bcc) lattice
    Three-dimensional lattice consisting of all points \((x,y,z)\) whose integer components are either all even or all odd.
  40. Bose–Chaudhuri–Hocquenghem (BCH) code[67]
    Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively prime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). More precisely, the generator polynomial of a BCH code of designed distance \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=q^r-1\) for some \(r\geq 2\). More general BCH codes can be defined for zeroes are powers of the form \(\{b,b+s,b+2s,\cdots,b+(\delta-2)s\}\) where gcd\((s,n)=1\).
  41. Bounded-energy code a.k.a. Spherical cluster.
    Code whose codewords are points lie on or inside a real or complex sphere whose radius squared is called the energy.
  42. Cameron-Goethals-Seidel (CGS) isotropic subspace code[68]
    Member of a \((q(q^2-q+1),(q+1)(q^3+1),2-2/q^2)\) family of spherical codes for any prime-power \(q\). Constructed from generalized quadrangles, which in this case correspond to sets of totally isotropic points and lines in the projective space \(PG_{5}(q)\) [60; Exam. 9.4.5]. There exist multiple distinct spherical codes using this construction for \(q>3\) [69].
  43. Cartier code[70]
    A generalization of the Goppa codes to codes defined from curves of non-zero genus. Each code is a subcode of a certain residue AG code and is constructed using the Cartier operator.
  44. Chien-Choy generalized BCH (GBCH) code[71]
    An \([n,k\geq n-rm, d\geq r+1]_q\) alternant code defined using two polynomials \(P(x),G(x)\) that are relatively prime to \(x^n-1\), with \(\deg P \leq n-1\) and \(r = \deg G \leq n-1\).
  45. Classical fractal liquid code[72,73]
    Member of a family of \([L^D,O(L^{D-1}),O(L^{D-\epsilon})]_p\) linear codes on \(D\)-dimensional square lattices of side length \(L\) and for some prime \(p\) and \(\epsilon > 0\) that is based on \(p\)-ary generalizations of the Sierpinski triangle.
  46. Classical topological code[74–76]
    Classical code defined on a two-dimensional lattice and derived from a geometrically local stabilizer code, such as the surface code or color code.
  47. Code in permutations[77,78] a.k.a. Permutation-based code.
    Encodes codewords into permutations of \(n\) objects.
  48. Code with locality
    A code with \((r,\delta)\) locality is a code that encodes each codeword coordinate into an \([r+\delta-1,r,\delta]\) MDS code [79; Sec. 31.3.4.5]. In other words, given a codeword \(c\) and coordinate \(c_i\), there exists a coordinate set \(S_i\) of size \(\leq r+\delta-1\) such that the restriction \(C_{|S_i}\) of the code to that set is a code with minimum distance \(\delta\).
  49. Combinatorial design a.k.a. Block design, Covering design.
    A constant-weight binary code that is mapped into a combinatorial \(t\)-design.
  50. Complete-intersection RM-type code[80]
    Evaluation code of polynomials evaluated on points lying on a complete intersection.
  51. Completely regular code[81]
    A code \(C\) is completely regular if the weight distribution of any coset \(e+C\) depends only on the distance \(d(e,C)\) of \(e\) to \(C\) [82].
  52. Complex Hadamard spherical code[83]
    A spherical code obtained from particular complex Hadamard matrices [84].
  53. Concatenated code[85] a.k.a. Serially concatenated code.
    A code whose encoding mapping is a composition of two mappings: first the message set is mapped onto the code space of the outer code, then each coordinate of the outer code is mapped on the code space of the inner code. In the basic construction, the outer code's alphabet is the finite field \(GF(p^m)\) and the \(m\)-dimensional inner code is over over the field \(GF(p)\). The construction is not limited to linear codes.
  54. Conference code[86][26; pg. 55]
    A member of the family of \((n,2n+2,(n-1)/2)\) nonlinear binary codes for \(n=1\) modulo 4 that are constructed from conference matrices.
  55. Constacyclic code a.k.a. Twisted code.
    A classical code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\). A \(-1\)-constacyclic code is called negacyclic.
  56. Constant-energy code
    Code whose codewords are points on a real or complex sphere whose radius squared is called the energy. Typically, only angular distances between points are relevant for code performance, so one can normalize codewords of a constant-energy code to obtain up a spherical code, i.e., a constant energy code with energy one.
  57. Constant-weight code
    A block code over a field or a ring whose codewords all have the same Hamming weight \(w\). The complement of a binary constant-weight code is a constant-weight code obtained by interchanging zeroes and ones in the codewords. The set of all binary codewords of length \(n\) forms the Johnson space \(J(n,w)\) [87–90].
  58. Constantin-Rao (CR) code[91]
    A nonlinear single-asymmetric-error code that generalize VT codes and that is constructed from an Abelian group.
  59. Construction-\(A\) code[92] a.k.a. Mod-two lattice.
    Sphere packing constructed from a binary \((n,K)\) code using Construction A [55].
  60. Convolutional code[93]
    Infinite-block code that is formed using generator polynomials over the finite field with two elements. The encoder slides across contiguous subsets of the input bit-string (like a convolutional neural network) evaluating the polynomials on that window to obtain a number of parity bits. These parity bits are the encoded information.
  61. Covering code
    A \(q\)-ary code \(C\) is \(\rho\)-covering if \(\forall v \in GF(q)^{n}\), there is a codeword \(c \in C\) such that the Hamming distance \(D(c,v)\leq \rho\). More generally, a covering code in a metric space is covering if the union of balls of some radius centered at the codewords covers the entire space.
  62. Coxeter-Todd \(K_{12}\) lattice[94]
    Even integral lattice in dimension \(12\) that exhibits optimal packing. It's automorphism group was discovered by Mitchell [95]. For more details, see [96][55; Sec. 4.9].
  63. Cross-interleaved RS (CIRS) code[97,98]
    An IRS code that is constructed using two shortened RS codes and two forms of interleaving. The code can also be visualized as a 2D array code [41].
  64. Cubeoctahedron code
    Spherical \((3,12,1)\) code whose codewords are the vertices of the cubeoctahedron. Codewords form the minimal lattice-shell code of the \(D_3\) face-centered cubic (fcc) lattice.
  65. Cycle LDPC code[99]
    An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.
  66. Cycle code[74,99–103] a.k.a. Graph theoretic code, Graph homology code, Graph code.
    A code whose parity-check matrix forms the incidence matrix of a graph. This code's properties are derived from the size two chain complex associated with the graph.
  67. Cyclic code[104–108]
    A code of length \(n\) over an alphabet is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword.
  68. Cyclic linear \(q\)-ary code
    A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.
  69. Cyclic linear binary code
    A binary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=2^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.
  70. Cyclic redundancy check (CRC) code[108–110] a.k.a. Frame check sequence (FCS).
    A generalization of the single parity-check code in which the generalization of the parity bit is the remainder of the data string (mapped into a polynomial via the Cyclic-to-polynomial correspondence) divided by some generator polynomial. A notable family of codes is referred to as CRC-(\(m-1\)), where \(m\) is the length of the generator polynomial.
  71. DNA storage code[111]
    Code that was designed (or that can be applied) to encode information into the four-base-pair alphabet of a DNA molecule.
  72. Deligne-Lusztig code[112–115]
    Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig curve.
  73. Delsarte-Goethals (DG) code[116]
    Member of a family of \((2^{2t+2},2^{(2t+1)(t-d+2)+2t+3},2^{2t+1}-2^{2t+1-d})\) binary nonlinear codes for parameters \(r \geq 1\) and \(m = 2t+2 \geq 4\), denoted by DG\((m,r)\), that generalizes the Kerdock code.
  74. Denniston code[117]
    Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{GF(2^a)}\) codes for \(i < a\) constructed using Denniston arcs.
  75. Determinant code[118]
    Determinant codes give optimal exact repair regenerating codes for any \([n,k,d=k]\) at all the points of the storage bandwidth trade-off curve. The codes are linear, and the exact regenerating property is provided based on fundamental properties of matrix determinants. The field size \(q\) required for this code construction is linear in \(n\).
  76. Diagonal code[119]
    Member of an explicit family of high-rate \([n,k,d,\alpha, \beta = \frac{\alpha}{d-k+1}, M=k\alpha]\) MSR codes for any \(r\) and \(n\). Such codes can optimally repair any \(f\) failed nodes from any \(d\) helper nodes for all \(d\), \(1 \le f \le r\) and \(k \le d \le n-f\) simultaneously. These codes can be constructed over any base field \(GF(q)\) as long as \(|GF(q)| \ge sn\), where \(s = \text{lcm}(1,2,\cdots,r)\).
  77. Difference-set cyclic (DSC) code[120]
    Cyclic LDPC code constructed deterministically from a difference set. Certain DCS codes satisfy more redundant constraints than Gallager codes and thus can outperform them [121].
  78. Dinur code[122]
    Member of infinite family of locally testable \([n,n/\text{polylog}(n),d]\) codes with vanishing rate. Code construction relies on a construction utilizing tensor-product codes [123].
  79. Disphenoidal 288-cell code
    Spherical \((4,48,2-\sqrt{2})\) code [60; Exam. 1.2.6] whose codewords are the vertices of the disphenoidal 288-cell. Codewords are the union of two 24-point lattice shells of the \(D_4\) lattice. The first shell consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\), and the second of the 16 vectors \((\pm 1,\pm 1,\pm 1,\pm 1)\) and the 8 permutations of the vectors \((0,0,0,\pm 2)\). A realization in terms of quaternion coordinates yields the 48 elements of the binary octahedral group \(2O\) [6; Sec. 8.6].
  80. Distributed computation code
    Encoding that provides an extra redundancy for distributed matrix computation algorithms such as matrix multiplication. Parallelized algorithms distribute a desired computation over many nodes, and a key performance bottleneck is due to some nodes completing their individual tasks much later than other nodes. Matrix computation codes provide a layer of redundancy such that the computation can be performed without having all nodes finish their piece of the computation.
  81. Distributed-storage code
    Block code designed to encode information into spatial nodes such that it is possible to recover said information after failure of some helper nodes by accessing the remaining nodes with minimal bandwidth.
  82. Divisible code[124]
    A linear \(q\)-ary block code is \(\Delta\)-divisible if the Hamming weight of each of its codewords is divisible by divisor \(\Delta\). A \(2\)-divisible (\(4\)-divisible, \(8\)-divisible) code is called even (doubly even, triply even) [55,125]. A code is called singly-even if all codewords are even and at least one has weight equal to 2 modulo 4. More generally, a code is \(m\)-even if it is \(2^{m}\)-divisible.
  83. Dodecacode[126]
    The unique trace-Hermitian self-dual additive \((12,4^6,6)_4\) code. Its codewords are cyclic permutations of \((\omega 10100100101)\), where \(GF(4)=\{0,1,\omega,\bar{\omega}\}\) is the quaternary Galois field [127; Sec. 2.4.8]. Another generator matrix can be found in [128; Exam. 9.10.8].
  84. Dodecahedron code
    Spherical \((3,20,2-2\sqrt{5}/3)\) code whose codewords are the vertices of the dodecahedron (alternatively, the centers of the faces of a icosahedron, the dodecahedron's dual polytope).
  85. Dual additive code
    For any \(q\)-ary additive code \(C\), the dual additive (or orthogonal additive) code is \begin{align} C^\perp = \{ y\in GF(q)^{n} ~|~ x \star y=0 \forall x\in C\}, \tag*{(2)}\end{align} where the trace inner product is \(x\star y = \sum_{i=1}^n \text{tr}(x_i y_i)\) for coordinates \(x_i,y_i\) and for \(\textit{tr}\) being the field trace.
  86. Dual code over \(R\)
    For any linear code \(C\) over a ring \(R\), the dual code over \(R\) is \begin{align} C^\perp = \{ y\in R^{n} ~|~ x \cdot y=0 \forall x\in C\}, \tag*{(3)}\end{align} where the ordinary, standard, or Euclidean inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\).
  87. Dual lattice a.k.a. Reciprocal lattice, Polar lattice.
    For any dimensional lattice \(L\), the dual lattice is the set of vectors whose inner products with the elements of \(L\) are integers.
  88. Dual linear code a.k.a. Orthogonal linear code.
    For any \([n,k]_q\) linear code \(C\), the dual code is the set of \(q\)-ary strings that are orthogonal to the codewords of \(C\) under a particular inner product.
  89. Dual polytope code
    For any spherical code whose codewords are vertices of a polytope, the dual code consists of codewords that are centers of the faces of said polytope. The dual codewords make up the vertices of the polytope dual to the original polytope.
  90. EVENODD code[129]
    A binary array code that can correct any two disk failures (i.e., two erasures). See [41] for more details.
  91. Editing code[130] a.k.a. Insertion and deletion code.
    A code designed to protect against insertions, where a new symbol is added somewhere within the string, and deletions, where a symbol at an unknown location is erased.
  92. Elliptic code
    Evaluation AG code of rational functions evaluated on points lying on an elliptic curve, i.e., a curve of genus one.
  93. Error-correcting code (ECC)
    Code designed for transmission of classical information through classical channels in a robust way.
  94. Error-correcting output code (ECOC)[131,132]
    A length-\(n\) binary or ternary (over \(\{\pm 1,0\}\)) block code used to convey information about classes to classifiers in multiclass machine learning. Rows of the code's generator matrix denote different classes, while columns correspond to classifiers. The \(\pm 1\) elements can be used to distinguish between a pair of chosen classes, while a zero entry correspond to a classifier ignoring that particular class.
  95. Evaluation AG code
    Evaluation code over \(GF(q)\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) whose corresponding vector space \(L\) of functions \(f\) consists of certain polynomials or rational functions.
  96. Evaluation code[133–135]
    Code whose codewords are evaluations of functions at certain fixed points. Code properties can be inferred from the structure of the functions and the underlying geometric object containing the points, often using results from algebraic geometry.
  97. Expander code[136] a.k.a. Sipser-Spielman code.
    LDPC code whose parity-check matrix is derived from the adjacency matrix of bipartite expander graph [137] such as a Ramanujan graph or a Cayley graph of a projective special linear group over a finite field [138,139]. Expander codes admit efficient encoding and decoding algorithms and yield an explicit (i.e., non-random) asymptotically good LDPC code family.
  98. Extended GRS code
    A GRS code with an additional parity-check coordinate with corresponding evaluation point of zero. In other words, an \([n+1,k,n-k+2]_q\) GRS code whose polynomials are evaluated at the points \((\alpha_1,\cdots,\alpha_n,0)\). The case when \(n=q-1\), multipliers \(v_i=1\), and \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is an extended narrow-sense RS code.
  99. Extended IRA (eIRA) code[140–142]
    A generalization of the IRA code in which the outer LDGM code is replaced by a random sparse matrix containing no weight-two columns.
  100. Fibonacci code[73]
    The code is defined on an \(L\times L/2\) lattice with one bit on each site, where \(L=2^N\) for an integer \(N\geq 2\). The codewords are defined to satisfy the condition that, for each lattice site \((x,y)\), the bits on \((x,y)\), \((x+1,y)\), \((x-1,y)\) and \((x,y+1)\) (where the lattice is taken to be periodic in both directions) contain an even numbers of \(1\)'s. The codewords can be generated using a one-dimensional cellular automaton of length \(L\) (periodic). The \(2^L\) possible initial states correspond to the \(2^L\) codewords. For each generation, the state of each cell is the xor sum of that cell and its two neighbors in the previous generation. After \(L/2-1\) generations, the entire history generated by the automaton corresponds to a codeword, where the initial state is the first row of the lattice, the first generation is the second row, etc.
  101. Finite-dimensional error-correcting code (ECC)[143]
    An error-correcting code defined over a finite alphabet.
  102. Finite-geometry LDPC (FG-LDPC) code[144]
    LDPC code whose parity-check matrix is the incidence matrix of points and hyperplanes in either a Euclidean or a projective geometry. Such codes are called Euclidean-geometry LDPC (EG-LDPC) and projective-geometry LDPC (PG-LDPC), respectively. Such constructions have been generalized to incidence matrices of hyperplanes of different dimensions [145].
  103. Flag-variety code[146]
    Evaluation code of polynomials evaluated on points lying on a flag variety.
  104. Folded RS (FRS) code[147]
    A linear \([n/m,k]_{q^m}\) code that is a modification of an \([n,k]_q\) RS code such that evaluations are grouped to yield a code with smaller length. In this case, the evaluation points are all powers of a generating field element \(\gamma\), \(\alpha_i=\gamma^i\). Each codeword \(\mu\) of an \(m\)-folded RS code is a string of \(n/m\) symbols, with each symbol being a string of values of a polynomial \(f_\mu\) at consecutive powers of \(\gamma\), \begin{align} \begin{split} \mu\to&\Big(\left(f_{\mu}(\alpha^{0}),\cdots,f_{\mu}(\alpha^{m-1})\right),\left(f_{\mu}(\alpha^{m}),\cdots,f_{\mu}(\alpha^{2m-1})\right)\cdots\\&\cdots,\left(f_{\mu}(\alpha^{n-m}),\cdots,f_{\mu}(\alpha^{n-1})\right)\Big)~. \end{split} \tag*{(4)}\end{align}
  105. Fountain code[148]
    Code based on the idea of generating an endless stream of custom encoded packets for the receiver. The code is designed so that the receiver can recover the original transmission of size \(Kl\) bits after receiving at least \(K\) packets each of \(l\) bits.
  106. Frameproof (FP) code[149,150]
    A block code designed to prevent a group of users from framing another user outside of the group for creating an unauthorized copy of data. FP codes help to provide software protection from the illegal distribution and copying of computer software and copyrighted materials. These codes help protect products of distributors as well as other naive users from being framed for illegal activity [151].
  107. Frequency-shift keying (FSK) code
    A \(q\)-ary frequency-shift keying (\(q\)-FSK) encodes one \(q\)-ary digit of information into signals with \(q\) different frequencies.
  108. Gabidulin code[152–154] a.k.a. Vector rank-metric code, Delsarte-Gabidulin code.
    A linear code over \(GF(q^N)\) that corrects errors over rank metric instead of the traditional Hamming distance. Every element \(GF(q^N)\) can be written as an \(N\)-dimensional vector with coefficients in \(GF(q)\), and the rank of a set of elements is rank of the matrix formed by their coefficients.
  109. Gallager (GL) code[155,156]
    The first LDPC code. The rows of the parity check matrix of this regular code are divided into equal subsets, and columns in the first subset are randomly permuted to yield the corresponding rows in subsequent subsets.
  110. Gauss' law code[157,158]
    An \([m+Dm,Dm,3]\) linear binary code for \(m\geq 3^D\), defined by the Gauss' law constraint of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory [158; Thm. 1]. The code can be re-phrased as a distance-one stabilizer code whose stabilizers consist of gauge-group elements. It can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [157,158].
  111. Generalized EVENODD code[159] a.k.a. Blaum-Bruck-Vardy array code.
    A generalization of the EVENODD code defined for parameters \(m\) and \(r\); see [41].
  112. Generalized Gallager code[160]
    A LDPC code that generalizes the Gallager codes using the Tanner construction. While Gallager code parity-check matrices consists of repetition code submatrices that are randomly permuted, generalized Gallager code matrices substitute general binary linear codes.
  113. Generalized RM (GRM) code[161–163]
    Reed-Muller code GRM\(_q(r,m)\) of length \(n=q^m\) over \(GF(q)\) with \(0\leq r\leq m(q-1)\). Its codewords are evaluations of the set of all degree-\(\leq r\) polynomials in \(m\) variables at the points of \(GF(q)\).
  114. Generalized RS (GRS) code
    An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors.
  115. Generalized Srivastava code[164]
    An \([n,k \geq n-mst,d \geq st+1 ]_q\) alternant code defined for \(n+s\) distinct elements \(\alpha_1,\alpha_2,\cdots,\alpha_n,w_1,w_2,\cdots,w_s\) and \(n\) nonzero elements \(z_1,z_2,\cdots,z_n\) of \(GF(q^m)\).
  116. Generalized concatenated code (GCC)[165,166] a.k.a. Cascade code.
    A code that combines multiple outer codes of the same length and (possibly) different dimensions with a single inner code; see Refs. [167][26; Ch. 18].
  117. Glynn code[168]
    The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
  118. Goethals code[169]
    Member of a family of \((2^m,2^{2^m-3m+1},8)\) binary nonlinear codes for \(m \geq 6\) that generalizes the Preparata codes. The code can be constructed as disjoint union of cosets of a certain linear code [26; Ch. 15].
  119. Golay code[170]
    A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [55] and sporadic simple groups [26]. Adding a parity bit to the code results in the self-dual \([24, 12, 8]\) extended Golay code. Up to equivalence, both codes are unique for their respective parameters [171]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [172]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [173,174].
  120. Gold code[175]
    Member of the family of \([2^r-1, 2r ]\) cyclic binary linear codes characterized by the generator polynomial of degree \(r\) of two maximum-period sequences of period \(2^r-1\) with absolute cross-correlation \( \leq 2^{(r+2)/2}\). Gold codewords are generated using \(m\)-sequences \(x\) and \(y\), which are codewords of simplex codes with check polynomials of degree \(r\) [175].
  121. Goldreich-Sudan code[176]
    Locally testable \([n,k,d]\) code with \(n = k^{1+O(1/u)}\) and distance of order \(\Omega(n)\) for query complexity \(u\). The same work also presented a probabilistic construction of codes of size \(k^{1+o(1)}\).
  122. Goppa code[177–179] a.k.a. LG code.
    Let \( G(x) \) be a polynomial describing a projective-plane curve with coefficients from \( GF(q^m) \) for some fixed integer \(m\). Let \( L \) be a finite subset of the extension field \( GF(q^m) \) where \(q\) is prime, meaning \( L = \{\alpha_1, \cdots, \alpha_n\} \) is a subset of nonzero elements of \( GF(q^m) \). A Goppa code \( \Gamma(L,G) \) is an \([n,k,d]_q\) linear code consisting of all vectors \(a = a_1, \cdots, a_n\) such that \( R_a(x) =0 \) modulo \(G(x)\), where \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \).
  123. Graph-adjacency code[180,181]
    Binary linear code whose generator matrix forms the adjacency matrix of a strongly regular graph. Given an adjacency matrix \(A\), the generator matrix is either \(G=A\) or \(G=(I|A)\), where \(I\) is the identity matrix.
  124. Grassmannian code[182–184]
    Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\) [185].
  125. Gray code[186–188]
    The first Gray code [186], now called the binary reflected Gray code, is a trivial \([n,n,1]\) code that orders length-\(n\) binary strings such that nearest-neighbor strings differ by only one digit.
  126. Griesmer code[189–191]
    A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality.
  127. Group-algebra code[192] a.k.a. \(G\) code.
    An \( [n,k]_q \) code whose automorphism group includes a finite group \( G \) of size \(n \), which acts on the code via its regular representation. This makes the code a \(G\)-submodule of the module \(GF(q)^n\) [194][193; Lemma 2.3]. A group-algebra code for an Abelian group is called an Abelian group-algebra code.
  128. Group-alphabet code
    Encodes \(K\) states (codewords) in coordinates labeled by elements of a group \(G\). The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice.
  129. Group-orbit code
    Code whose set of codewords forms an orbit of some reference codeword under a subgroup of the automorphism group, i.e., the group of distance-preserving transformations on the metric space defined with the code's alphabet.
  130. Hadamard code a.k.a. Walsh code, Walsh-Hadamard code.
    An \([2^m,m,2^{m-1}]\) balanced binary code. The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)), while the \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).
  131. Hansen toric code[195,196]
    Evaluation code of a linear space of polynomials evaluated on points lying on an affine or projective toric variety. If the space is taken to be all polynomials up to some degree, the code is called a toric RM-type code of that degree.
  132. Hergert code[116] a.k.a. Goethals-Delsarte (GD) code.
    A nonlinear subcode of an RM code that is a formal dual of the nonlinear DG code in the sense that its distance distribution is equal to the MacWilliams transform of the distance distribution of a DG codes.
  133. Hermitian code[197,198][134; Sec. 4.4.3]
    Evaluation AG code of rational functions evaluated on points lying on a Hermitian curve in either affine or projective space. Hermitian codes improve over RS codes in length: that RS codes have length at most \(q+1\) while Hermitian codes have length \(q^3 + 1\).
  134. Hermitian-hypersurface code[199]
    Evaluation code of polynomials evaluated on points lying on a Hermitian hypersurface.
  135. Hessian polyhedron code[200,201] a.k.a. Schläfli configuration.
    Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [60; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [69,202–204].
  136. Hexacode[55,205]
    The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [55], and conformal field theory [206]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [207]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).
  137. Higman-Sims graph-adjacency code[180,181]
    A graph-based code whose generator matrix is constructed using the adjacency matrix \(A\) of the Higman-Sims graph. Setting the generator matrix \(G=(I|A)\) yields a \([100,22,32]\) code whose dual is an optimal \([100,78,8]\) code [180; Table VI].
  138. Hill projective-cap code[208]
    Member of a projective code family that contains of \(q\)-ary sharp configurations and that is constructed using projective caps.
  139. Hirschfeld code[209]
    A projective geometry code that is an example of an MDS code that is not an RS code; see [210; Exam. 7.6] for the description.
  140. Hoffman-Singleton cycle code[180,181]
    A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [211]. Its dual is a \([50,29,8]\) code [180; Table II].
  141. Hoffman-Singleton graph-adjacency code[180,181]
    A graph-based code whose generator matrix is constructed using the adjacency matrix of the Hoffman-Singleton graph [211]. Setting the generator matrix equal to the adjacency matrix, \(G=A\), yields a \([50,22,7]\) code whose dual is a \([50,28,8]\) code [180; Table III].
  142. Honeycomb tiling a.k.a. Hexagonal tiling.
    A two-dimensional point set whose points are vertices of hexagons. It is not a lattice since its points do not form a group under addition. As a tiling, its dual (whose points lie at the centers of each triangle) is the triangular tiling. The ruby tiling is a fattened honeycomb tiling interpolating between the honeycomb tiling and triangular lattice.
  143. Hsu-Anastasopoulos LDPC (HA-LDPC) code[212]
    A regular LDPC code obtained from a concatenation of a certain random regular LDPC code and a certain random LDGM code. Using rate-one LDGM codes eliminates high-weight codewords while admitting an amount of low-weight codewords that asymptotically vanishes, allowing code families to achieve the GV bound with high probability.
  144. Hyperbolic evaluation code[213–215] a.k.a. Hyperbolic cascaded RS code.
    An evaluation code over polynomials in two variables. Generator matrices are determined in Ref. [215] following initial formulations of the codes as generalized concatenations of RS codes [213,214]; see [133; Exam. 4.26].
  145. Hyperbolic sphere packing[216,217]
    Encodes states (codewords) into coordinates in the hyperbolic plane \(\mathbb{H}^2\).
  146. Hypercube code
    Spherical \((n,2^n,4/n)\) code whose codewords are vertices of an \(n\)-cube, i.e., all permutations and negations of the vector \((1,1,\cdots,1)\), up to normalization.
  147. Hyperoval code[218]
    A projective code constructed using hyperovals in projective space.
  148. Icosahedron code
    Spherical \((3,12,2-2/\sqrt{5})\) code whose codewords are the vertices of the icosahedron (alternatively, the centers of the faces of a dodecahedron, the icosahedron's dual polytope).
  149. Identifiable parent property (IPP) code[219]
    A code that is embedded in copyrighted content in order to detect unauthorized redistribution of said content by pirates. IPP codes are designed to detect pirates even when segments content are mixed together so as to conceal the pirates' identities.
  150. Incidence-matrix projective code[220–222]
    Code whose generator matrix is the incidence matrix of points and hyperplanes in a projective space. Has been generalized to incidence matrices of other structures [223,224][225; Sec. 14.4]. Columns of a code's parity-check matrix can similarly correspond to an incidence matrix.
  151. Interleaved RS (IRS) code
    A modification of RS codes where multiple polynomials are used to define each codeword. Each codeword \(\mu\) of a \(t\)-interleaved RS code is a string of values of the corresponding set \(\{f_\mu^{(1)},f_\mu^{(2)},\cdots,f_\mu^{(t)}\}\) of \(t\) polynomials at the points \(\alpha_i\). The vector codewords can be arranged in an array whose rows are ordinary RS codes for each polynomial \(f^{j}\), yielding the encoding \begin{align} \mu\to\left( \begin{array}{cccc} f_{\mu}^{(1)}\left(\alpha_{1}\right) & f_{\mu}^{(1)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(1)}\left(\alpha_{n}\right)\\ f_{\mu}^{(2)}\left(\alpha_{1}\right) & f_{\mu}^{(2)}\left(\alpha_{2}\right) & & f_{\mu}^{(2)}\left(\alpha_{n}\right)\\ \vdots & & \ddots & \vdots\\ f_{\mu}^{(t)}\left(\alpha_{1}\right) & f_{\mu}^{(t)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(t)}\left(\alpha_{n}\right) \end{array}\right)~. \tag*{(5)}\end{align}
  152. Irregular LDPC code[226,227]
    An LDPC code whose parity-check matrix has a variable number of entries in each row or column.
  153. Irregular repeat-accumulate (IRA) code[228–230]
    A generalization of the RA code in which the outer 1-in-3 repetition encoding step is replaced by an LDGM code. A simple version is when different bits in the RA block are repeated a different number of times.
  154. Julin-Golay code[54,231,232]
    One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [233,234][26; Sec. 2.7][235; Sec. 4] or their shortened versions, the nonlinear \((11,72,4)\), \((10,38,4)\), and \((9,20,4)\) Julin-Golay codes. Several of these codes contain more codewords than linear codes of the same length and distance and yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.
  155. Justesen code[236]
    Binary linear code resulting from generalized concatenation of an outer RS code with multiple inner codes sampled from the Wozencraft ensemble, i.e., \(N\) distinct binary inner codes of dimension \(m\) and length \(2m\). The first asymptotically good codes.
  156. Kasami code[237]
    Member of the family of \([2^{2r}-1, 3r, 2^{2r-1} - 2^{r-1} ]\) cyclic binary linear codes.
  157. Kerdock code[238]
    Binary nonlinear \((2^m, 2^{2m}, 2^{m-1} - 2^{(m-2)/2})\) for even \(m\) consisting of the first-order Reed-Muller code RM\((1,m)\) with maximum-rank cosets of RM\((1,m)\) in RM\((2,m)\).
  158. Kerdock spherical code[239–241]
    Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\), obtained from Kerdock codes via the antipodal mapping [60; pg. 157]. These codes are optimal for their parameters for \(2\leq r\leq 5\), they are unique for \(r\in\{2,3\}\), and they form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [242].
  159. Klein-quartic code[243]
    Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) ([133], Exam. 2.75).
  160. Kopparty-Meir-Ron-Zewi-Saraf (KMRS) code[244,245]
    Member of a family of locally testable binary linear codes with constant rate, constant relative distance, and subpolynomial query complexity \(u = (\log n)^{O(\log \log n)}\)). Later work by Gopi, Kopparty, Oliveira, Ron-Zewi, and Saraf [245] showed that related concatenated codes achieve the GV bound.
  161. LDPC convolutional code (LDPC-CC)[246–248] a.k.a. Low-density convolutional (LDC) code.
    Convolutional code defined by an infinite low-density parity-check matrix.
  162. Laminated spherical code[249]
    Spherical code whose codewords are obtained from a recursive procedure that is similar to the procedure that creates laminated lattices.
  163. Lattice-based code
    Encodes states (codewords) in coordinates of an \(n\)-dimensional lattice, i.e., a discrete set of points in Euclidean space \(\mathbb{R}^n\) that forms a group under vector addition when the set is translated such that one point is at the origin. The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals.
  164. Lattice-shell code[250,251]
    Spherical code whose codewords are scaled versions of points on a lattice. A \(m\)-shell code consists of normalized lattice vectors \(x\) with squared norm \(\|x\|^2 = m\). Each code is constructed by normalizing a set of lattice vectors in one or more shells, i.e., sets of lattice points lying on a hypersphere.
  165. Lazebnik-Ustimenko (LU) code[252,253]
    LDPC code whose Tanner graph comes from a particular family of \(q\)-regular graphs [252] of known girth and relatively large stopping sets.
  166. Left-right Cayley complex code[254]
    Binary code constructed on a left-right Cayley complex using a pair of base codes \(C_A,C_B\) and an expander graph [137] such that codewords for a fixed graph vertex are codewords of the tensor code \(C_A \otimes C_B\). A family of such codes is one of the first \(c^3\)-LTCs.
  167. Levenshtein code[255]
    Binary codes constructed from combining two codes \(A'\) constructed out of Hadamard matrices.
  168. Lexicographic code[256,257]
    A \(q\)-ary code whose codewords are constructed greedily and iteratively by starting with zero and adding codewords whose distance is the desired minimum distance of the code.
  169. Linear STC
    Spacetime code whose set of matrix codewords is closed under addition and subtraction.
  170. Linear \(q\)-ary code
    An \((n,K,d)_q\) linear code is denoted as \([n,k,d]_q\), where \(k=\log_q K\) need not be an integer. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(\alpha x+ \beta y\) is also a codeword for any \(q\)-ary digits \(\alpha,\beta\). This extra structure yields much information about their properties, making them a large and well-studied subset of codes.
  171. Linear binary code
    An \((n,2^k,d)\) linear code is denoted as \([n,k]\) or \([n,k,d]\), where \(d\) is the code's distance. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword. A code that is not linear is called nonlinear.
  172. Linear code over \(G\)[258–260]
    Block code that encodes \(K\) states (codewords) in \(n\) coordinates over a group \(G\) such that the codewords form a subgroup of \(G^n\). In other words, the set of codewords is closed under the group operation.
  173. Linear code with complementary dual (LCD)[261]
    A linear code \(C\) admits a complementary dual if \(C\) and its dual code \(C^{\perp}\) do not share any codewords.
  174. Linearized RS code[262–264]
    A linearized version of a skew RS code, i.e., an RS code constructed by evaluating skew polynomials [262,263].
  175. Locally correctable code (LCC)
    Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word. Informally, an LCC is a block code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)).
  176. Locally decodable code (LDC)[265]
    Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word. Informally, an LDC is a block code for which one can recover any coordinate of the message from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)). Efficiency of the correction is quantified by the code's query complexity \(r\), and correction is performed by sampling subsets of \(r\) bits.
  177. Locally recoverable code (LRC) a.k.a. Locally repairable code.
    An LRC is a block code for which one can recover any coordinate of a codeword from at most \(r\) other coordinates of the codeword. In other words, an LRC of locality \(r\) is a block code for which, given a codeword \(c\) and coordinate \(c_i\), \(c_i\) can be recovered from at most \(r\) other coordinates of \(c\). An \(r\)-locally recoverable code of length \(n\) and dimension \(k\) is denoted as an \((n,k,r)\) LRC. The definition can be generalized to \(t\)-LRC, if every coordinate is recoverable from \(t\) disjoint subsets of coordinates.
  178. Locally testable code (LTC)[266–269]
    Code for which one can efficiently check whether a given string is a codeword or is far from a codeword. Efficiency of the verification is quantified by the code's query complexity \(u\), while effectiveness is quantified by the code's soundness \(R\).
  179. Long code[270,271]
    Locally testable \([2^{2^k},k,d]\) code. The encoder maps a \(k\)-bit string into a codeword that consists of the values of all Boolean functions on the \(k\)-bit string. The code is not practical, but is useful for certain probabilistically checkable proof (PCP) constructions [272].
  180. Low-density generator-matrix (LDGM) code
    Binary linear code with a sparse generator matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the generator matrix are both bounded by a constant as \(n\to\infty\). The dual of an LDGM code has a sparse parity-check matrix and is called an LDPC code.
  181. Low-density parity-check (LDPC) code[155,156] a.k.a. Sparse graph code.
    A binary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
  182. Low-rank parity-check (LRPC) code[273]
    An LRPC code of rank \(d\) is a rank-metric code that, when interpreted as a linear code over \(GF(q^m)\), admits an \((n-k)\times n\) parity-check matrix whose entries span a subspace of \(GF(q^m)\) that is at most \(d\)-dimensional.
  183. Luby transform (LT) code[274]
    Erasure codes based on fountain codes. They improve on random linear fountain codes by having a much more efficient encoding and decoding algorithm.
  184. MDS array code
    An \((n,k,m)\) array code whose codewords can be recovered by any \(k\) out of \(n\) nodes, where each node stores a length-\(m\) column of the codeword. MDS array codes are MDS codes when each matrix codeword is treated as a vector by converting each column into a single coordinate via subpacketization.
  185. MacKay-Neal LDPC (MN-LDPC) code[275,276]
    Codes whose parity-check matrix is constructed non-deterministically via the MacKay-Neal prescription. The parity-check matrix of an \((l,r,g\))-MN-LDPC code is of the form \((H_1~H_2)\), where \(H_1\) is a random binary matrix of column weight \(l\) and row weight \(r\), and \(H_2\) is a random binary matrix of column and row weight \(g\) [277].
  186. Margulis LDPC code[9]
    Member of a class of LDPC codes deterministically constructed using a class of regular graphs with no short cycles. Related explicit LDPC constructions [278] utilize Ramanujan graphs [138,139].
  187. Matrix-based code a.k.a. Two-dimensional code.
    Encodes \(K\) states (codewords) in an \(m\times n\)-dimensional matrix of coordinates over a field (e.g., the Galois field \(GF(q)\) or the complex numbers \(\mathbb{C}\)).
  188. Matrix-product code[279]
    Code constructed using a concatenation procedure that yields a code consisting of all products of codewords in \(M\) length-\(n\) \(q\)-ary codes and an \(M\times N\) \(q\)-ary matrix with \(N\geq M\). A prominent subclass is the case with \(A\) is non-singular by columns (NSC).
  189. Maximally recoverable (MR) code[280,281] a.k.a. Partial MDS code.
    A code with \((r,\delta)\) locality such that puncturing it on any \(\delta-1\) coordinates of the local \([r+\delta-1,r,\delta]\) codes yields an MDS code.
  190. Maximum distance separable (MDS) code[282]
    A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality.
  191. Maximum-rank distance (MRD) code[153,154,283] a.k.a. Optimal rank-distance code.
    An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the Singleton-like bound \begin{align} k \leq \max(n, m) (\min(n, m) - d + 1) \tag*{(6)}\end{align} becomes an equality.
  192. Maximum-sum-rank distance (MSRD) code[264] a.k.a. Optimal sum-rank-distance code.
    An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the sum-rank-metric Singleton bound [264; Prop. 34] \begin{align} d_{\text{SR}}(C) \leq n - k + 1 \tag*{(7)}\end{align} becomes an equality, where \(d_{\text{SR}}\) is the sum-rank metric.
  193. Meir code[284]
    Locally testable \([n,k,d]_q\) code with query complexity \(\text{poly}(\log k)\) and rejection ratio \(R/n = 1/\text{poly}(\log k)\). Code construction is probabilistic and combinatorial.
  194. Melas code[285,286]
    Cyclic \([2^m -1, 2^m - 1 - 2m, 5]\) linear code with generator polynomial is \(g(x) = p(x)p(x)^{\star}\), where \(p(x)\) is a primitive polynomial of degree \(m\) that is the minimal polynomial over \(GF(2)\) of an element \(\alpha\) of order \(2^m -1\) in \(GF(2^m)\), \(m\) is odd and greater that five, and '\(\star\)' denotes reciprocation [287].
  195. Minimum-bandwidth regenerating (MBR) code
    An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = d\beta\).
  196. Minimum-storage regenerating (MSR) code
    An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = (d-k+1)\beta\).
  197. Mixed code a.k.a. Mixed-alphabet code.
    Encodes \(K\) states (codewords) in a string of two or more coordinates, each of which takes values in one of two or more possible groups.
  198. Modulation scheme
    A sphere packing mapped into a time-dependent electromagnetic signal [288,289]. There is a close relation between abstract real-space encodings and modulation schemes, and certain simple sphere packings are often synonymous with their corresponding modulation schemes.
  199. Multi-channel group-orbit code[290]
    Extension of binary group-orbit codes to multi-antenna transmission.
  200. Multi-edge LDPC code[291]
    Irregular LDPC code whose construction generalizes those of the original examples of irregular LDPC as well as RA codes.
  201. Multiplicity code[292–294]
    A generalization of an \(m\)-variate polynomial evaluation code based on evaluating polynomials and \(s\) of their derivatives at all points in \(GF(q)^m\). Originally proposed for coding using the Rosenbloom-Tsfasman metric [292]. Univariate (\(m=1\)) [292,293] and multivariante (\(m>1\)) [294] codes have been proposed.
  202. Nadler code[295]
    A nonlinear \((12,32,5)\) binary code that is the largest double-error-correcting code.
  203. Narrow-sense RS code[205,296,297]
    An \([q-1,k,n-k+1]_q\) RS code whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element \(\alpha\) of \(GF(q)\).
  204. Nearly perfect code[298–300]
    A type of binary code whose parameters satisfy the Johnson bound with equality.
  205. Newman-Moore code[301]
    Member of a family of \([L^2,O(L),O(L^{\frac{\log 3}{\log 2}})]\) binary linear codes on \(L\times L\) square lattices that form the ground-state subspace of a class of exactly solvable spin-glass models with three-body interactions. The codewords resemble the Sierpinski triangle on a square lattice, which can be generated by a cellular automaton [302].
  206. Niederreiter-Rosenbloom-Tsfasman (NRT) code[292,303–305]
    A poset code based on the total ordering of \([n]\), i.e., \(1\leq 2\leq \cdots \leq n\).
  207. Niemeier lattice[306]
    One of the 24 positive-definite even unimodular lattices of rank 24.
  208. Nonlinear AG code[307–311]
    Nonlinear \(q\)-ary code constructed by evaluating functions on an algebraic curve.
  209. Nordstrom-Robinson (NR) code[312,313]
    A nonlinear \((16,256,6)\) binary code that is the smallest Kerdock and the smallest Preparata code. The size of this code is larger than the largest possible linear code with the same length and distance.
  210. Norm-trace code[314]
    Evaluation AG code of rational functions evaluated on points lying on a Miura-Kamiya curve in either affine or projective space. The family is named as such because the equations defining the curves can be expressed in terms of the field norm and field trace.
  211. Octacode[55,315,316]
    The unique self-dual linear code of length 8 and Lee distance 6 over \(\mathbb{Z}_4\) with generator matrix \begin{align} \begin{pmatrix} 3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\ 3 & 0 & 3 & 2 & 3 & 1 & 0 & 0\\ 3 & 0 & 0 & 3 & 2 & 3 & 1 & 0\\ 3 & 0 & 0 & 0 & 3 & 2 & 3 & 1 \end{pmatrix}\,. \tag*{(8)}\end{align}
  212. One-hot code[317] a.k.a. One-vs-all (OVA) code, One-against-all (1AA) code, One-vs-rest (OvR) code, \(1\)-in-\(n\) code.
    A length-\(n\) binary code whose codewords are those with Hamming weight one. The reverse of this code, where all codewords have Hamming weight \(n-1\) is called a one-cold code.
  213. One-versus-one (OVO) code[318,319] a.k.a. One-against-one (1A1) code.
    A length-\(n\) ternary code over \(\{\pm 1,0\}\) whose whose generator matrix has columns with one \(+1\), one \(-1\), and with the rest of the entries zero.
  214. Optimal LRC[320,321]
    An LRC whose parameters saturate a generalized Singleton bound.
  215. Orthogonal Spacetime Block Code (OSTBC)[8]
    The codewords are \(T\times n\) matrices as defined for spacetime codes, with the additional condition that columns of the coding matrix are orthogonal. The parameter \(n\) is the number of channels, and \(T\) is the number of time slots.
  216. Orthogonal array (OA)[322–324]
    An orthogonal array, or OA\(_{\lambda}(t,n,q)\), of strength \(t\) with \(q\) levels and \(n\) constraints is a set of \(q\)-ary strings such that any subset of \(t\) coordinates contains every length-\(t\) string an equal number of times \(\lambda\), which is the index of the array.
  217. Ovoid code[218,325]
    Member of a \([q^2+1,4,q^2-q]_q\) projective code family that is universally optimal and that is constructed using ovoids in projective space. See [326; pg. 107][32; pg. 192] for further details.
  218. Parallel concatenated code
    A code that is constructed by combining two or more codes in a Tanner code, in a tensor-product code, or in a modified Tanner construction [327].
  219. Parallel-recovery code[328]
    A \(t\)-erasure LRC whose coordinate erasures can be recovered in parallel.
  220. Parvaresh-Vardy (PV) code[329] a.k.a. Correlated RS code.
    An IRS code with additional algebraic relations (a.k.a. correlations) between the codeword polynomials \(\{f^{(j)}\}_{j=1}^{t}\). These relations yielded a list decoder that achieves list-decoding capacity.
  221. Pentacode[330]
    Nonlinear \((5,40,4)_{\mathbb{Z}_4}\) code over \(\mathbb{Z}_4\) whose codewords are all cyclic permutations and negations of the strings \(01112\), \(03110\), \(21310\), and \(21132\).
  222. Pentakis dodecahedron code
    Spherical \((3,32,(9-\sqrt{5})/6)\) code whose codewords are the vertices of the pentakis dodecahedron, the convex hull of the icosahedron and dodecahedron.
  223. Perfect binary code
    An \((n,K,2t+1)\) binary code is perfect if parameters \(n\), \(K\), and \(t\) are such that the binary Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t} {n \choose j} \leq 2^{n}/K \tag*{(9)}\end{align} becomes an equality. For example, for a code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) binary strings.
  224. Perfect code
    A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality.
  225. Permutation spherical code[331,332]
    Slepian group-orbit code whose codewords are constructed from an arbitrary unit vector in two possible variants. Variant 1 consists of codewords that are permutations of the vector's coordinates, while Variant 2 consists of such permutations and all possible sign changes of the vector's components.
  226. Petersen cycle code[99]
    A \([15,6,5]\) cycle code whose parity-check matrix is the incidence matrix of the Petersen graph. The Petersen graph can be thought of as a dodecahedron with antipodes identified [333; Appx. A.2.1].
  227. Petersen spherical code[334]
    A \((4,10,1/6)\) spherical code whose codewords correspond to vertices of the Peterson graph. Its Gram matrix is constructed by putting \(-2/3\) whenever two vertices are adjacent in the graph, and \(1/6\) otherwise. The code is optimal for its parameters [334].
  228. Phase-shift keying (PSK) code
    A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\).
  229. Plane-curve code[335]
    Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve.
  230. Pless symmetry code[336,337] a.k.a. Pless double circulant code.
    A member of a family of self-dual ternary \([2q+2,q+1]_3\) codes for any power of an odd prime satisfying \(q \equiv 2\) modulo 3.
  231. Polar code[338]
    In its basic version, a binary linear polar code encodes \(K\) message bits into \(N=2^n\) bits. The linear transformation that defines the code is given by the matrix \(G^{(n)}=B_N G^{\otimes n}\), where \(B_N\) is a certain \(N\times N\) permutation matrix, and \(G^{\otimes n}\) is the \(n\)th Kronecker power of the \(2\times 2\) kernel matrix \(G=\left[\begin{smallmatrix}1 & 0\\ 1 & 1 \end{smallmatrix}\right]\). To encode \(K\) message bits, one forms an \(N\)-vector \(u\) in which \(K\) coordinates represent the message bits. The remaining \(N-K\) coordinates are set to some fixed values and are said to be frozen. The codeword \(x \in \{0,1\}^N\) is obtained as \(x=u G^{\otimes n}\).
  232. Polygon code
    Spherical \((1,q,4\sin^2 \frac{\pi}{q})\) code for any \(q\geq1\) whose codewords are the vertices of a \(q\)-gon. Special cases include the line segment (\(q=2\)), triangle (\(q=3\)), square (\(q=4\)), pentagon (\(q=5\)), and hexagon (\(q=6\)).
  233. Polyhedron code
    A polytope code in three dimensions, i.e., a spherical code whose codewords form vertices of a polyhedron.
  234. Polynomial evaluation code
    Evaluation code of polynomials (or, more generally, rational functions) at points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) on an algebraic variety \(\cal X\) of dimension greater than one (i.e., not an algebraic curve).
  235. Polyphase code[339–349]
    A spherical code obtained from a binary code, \(q\)-ary code, or \(q\)-ary code over \(\mathbb{Z}_q\) via a component-wise mapping of each \(q\)-ary digit to a \(q\)th root of unity in a generalization of the antipodal mapping.
  236. Polytope code
    Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes in either real [3] or complex [350] space. A polytope in two (three, four) dimensions is called a polygon (polyhedron, polychoron).
  237. Poset code[351]
    Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), with its distance evaluated in the poset metric.
  238. Preparata code[352]
    A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd. The size of this code is twice the size of the largest possible linear code with the same length and distance.
  239. Primitive narrow-sense BCH code[67]
    BCH codes for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\).
  240. Private information retrieval (PIR) code[353,354]
    A code used to obtain information from several servers privately, i.e., without the servers knowing what information was obtained.
  241. Product-matrix (PM) code[355]
    Code constructed using two explicit constructions, with each construction corresponding to one of the two extreme points of the storage-bandwidth trade-off curve [356].
  242. Projective RM (PRM) code[357,358]
    Reed-Muller code for nonzero points \(\{\alpha_1,\cdots,\alpha_n\}\) with \(n=m+1\) whose leftmost nonzero coordinate is one, corresponding to an evaluation code of polynomials over projective coordinates.
  243. Projective geometry code
    Linear \(q\)-ary \([n,k,d]\) code such that columns of its generator matrix \(G\) does not contain any repeated columns or the zero column. That way, each column corresponds to a distinct point in the projective space \(PG(k-1,q)\) arising from a \(k\)-dimensional vector space over \(GF(q)\). If the columns are linearly independent, then the codewords are collectively called an information set. Columns of a code's parity-check matrix can similarly correspond to points in projective space. This formulation yields connections to projective geometry, which can be applied to determine code properties.
  244. Projective two-weight code
    A projective code whose codewords all have one of two possible nonzero Hamming weights.
  245. Protograph LDPC code[359–361]
    Binary version of a \(q\)-ary protograph LDPC code. Its parity check matrix can be put into the form of a block matrix consisting of either a sum of permutation sub-matrices or the zero sub-matrix.
  246. Pulse-amplitude modulation (PAM) code
    Encodes a \(q\)-ary digit into a constellation of equally spaced points on the real line. For example, a \(q\)-PAM scheme for \(q=8\) could encode the constellation \(\{ \pm \alpha,\pm 3\alpha,\pm 5\alpha, \pm 7\alpha \}\) with real scaling factor \(\alpha\). The points in the constellation are typically associated with one quadrature of an electromagnetic signal.
  247. Pulse-position modulation (PPM) code
    An analog code encoding into \(q\) different signals such that each codeword corresponds to a signal.
  248. Pyramid code[362]
    An LRC whose generator matrix is that of an RS code in standard form, but some of whose columns are split into multiple columns; see [79; Sec. 31.3.1.1] for an example.
  249. Quadrature PSK (QPSK) code[363] a.k.a. Quadriphase PSK code, 4-PSK code, 4-QAM code.
    A quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\).
  250. Quadrature-amplitude modulation (QAM) code
    Encodes into points into a subset of points lying on in \(\mathbb{R}^{2}\), here treated as \(\mathbb{C}\). Each pair of points is associated with a complex amplitude of an electromagnetic signal, and information is encoded into both the norm and phase of that signal [289; Ch. 16].
  251. Quadric code[364,365]
    Evaluation code of polynomials evaluated on points lying on a quadric hypersurface.
  252. Quantum-inspired classical block code
    A block code of length \(n\) whose construction was inspired by a quantum code.
  253. Quasi group-algebra code a.k.a. Quasi-\(G\) code.
    A \(q\)-ary linear code based on a finite group \( G \) of order \(n/\ell\) for some index \(\ell\). The code is a right submodule of the direct sum of \(\ell\) copies of the group algebra \(\mathbb{F}_q G\). A quasi group-algebra code for an Abelian group is called an Abelian quasi group-algebra code.
  254. Quasi-cyclic LDPC (QC-LDPC) code[18,366–371][156; Appx. C]
    LDPC code that can be put into quasi-cyclic form. Its parity check matrix can be put into the form of a block matrix consisting of either circulant permutation sub-matrices or the zero sub-matrix. Such codes are often constructed by lifting certain protographs into such block matrices [372]. Their simple structure makes them useful for several wireless communication standards.
  255. Quasi-cyclic code[373]
    A block code of length \(n\) is quasi-cyclic if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(c_{n-\ell+1} \cdots c_n c_1 \cdots c_{n-\ell}\), where each entry is cyclically shifted by \(\ell\) increments, is also a codeword.
  256. Quasi-perfect code
    Perfect codes \((n,K,d)_q\) are those for which balls of Hamming radius \(t=\left\lfloor (d-1)/2\right\rfloor\) exactly fill the space of all \(n\) \(q\)-ary strings. Quasi-perfect codes are those for which balls of Hamming radius \(t\) are disjoint, while balls of radius \(t+1\) cover the space with possible overlaps. In other words, any \(q\)-ary string is at most \(t+1\) bit flips away from a codeword of a quasi-perfect code.
  257. Quasi-twisted code
    A block code of length \(n\) is \(\alpha\)-quasi-twisted if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(\alpha c_{n-\ell+1}, \alpha c_{n-\ell+2}, \cdots, \alpha c_n, c_1, c_2, \cdots, c_{n-\ell}\) is also a codeword.
  258. Quaternary RM (QRM) code[374]
    A quaternary linear code over \(\mathbb{Z}_4\) that is a quaternary version of the RM code in that its binary image under the Gray map is an RM code. This code subsumes the quaternary images of the Kerdock and Preparata codes under the Gray map. The code is usually noted as QRM\((r,m)\), with its image under the Gray map yielding the RM code RM\((r,m)\) [374; Thm. 19].
  259. Quaternary linear code over \(\mathbb{Z}_4\)
    A linear code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_4\) of integers modulo 4.
  260. RS NRT code[292]
    An NRT analogue of an RS code.
  261. Random code[143]
    Code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes, to codes whose multi-step construction is deterministic with the exception of a single step.
  262. Rank-metric code[153] a.k.a. Delsarte rank-metric code.
    Each codeword is a matrix over \(GF(q)\), with codewords forming a \(GF(q)\)-linear subspace, and with the metric being the rank of the difference of matrices. The distance \(d\) is the minimum rank of all nonzero matrices in the code. Rank-metric codes on \(n\times m\) matrices are denoted as \([n\times m,k,d]_q\).
  263. Rank-modulation code[375,376]
    A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes [376], quadratic residue codes, and most binary codes.
  264. Raptor (RAPid TORnado) code[377,378]
    Raptor codes are concatenated erasure codes with two layers: an outer pre-code and a Luby-Transform (LT) inner code. The pre-code is a linear binary erasure code, which is applied first to the input to create some redundant data. The LT code is then applied to the intermediate symbols from the pre-code to generate final output symbols to be transmitted.
  265. Real-Clifford subgroup-orbit code[379,380]
    Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\). Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [381], onto the vector \((1,0,0,\cdots,0)\). This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\).
  266. Rectified Hessian polyhedron code
    Spherical \((6,72,1)\) code whose codewords are the vertices of the rectified Hessian complex polyhedron and the \(1_{22}\) real polytope. Codewords form the minimal lattice-shell code of the \(E_6\) lattice. See [350; pg. 127][55; pg. 126] for realizations of the 72 codewords.
  267. Reed-Muller (RM) code[382–384]
    Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the order of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called universe codes. Punctured RM codes RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword.
  268. Reed-Solomon (RS) code[205,296,297]
    An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\).
  269. Regenerating code (RGC)[356]
    An \([n,k,d,\alpha,\beta,M]_q\) Regenerating Code \(\mathcal{C}\) is an erasure correcting code that encodes \(M\) symbols from \(GF(q)\) into an \(\alpha \times n\) matrix over \(GF(q)\), with each column of the matrix treated as a coordinate of a codeword.
  270. Regular LDPC code
    An LDPC code whose parity-check matrix has a fixed number of entries for each row or column.
  271. Regular binary Tanner code[385] a.k.a. Regular binary GLDPC code.
    A binary Tanner code defined on a regular bipartite graph, with the inner code being the same for all vertices.
  272. Repeat-accumulate (RA) code[386]
    An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [387].
  273. Repeat-accumulate-accumulate (RAA) code[388]
    Generalization of the RA code in which two accumulators and permutations are used.
  274. Repetition code
    \([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. Its automorphism group is \(S_n\).
  275. Residue AG code a.k.a. Differential code.
    Linear \(q\)-ary code defined using a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) and a linear space \(\Omega\) of certain rational differential forms \(\omega\).
  276. Reversible code
    A code of length \(n\) over an alphabet is reversible if, for each codeword \(c_1 c_2 \cdots c_n\), the reversed string \(c_n \cdots c_2 c_1\) is also a codeword.
  277. Ring code
    Encodes \(K\) states (codewords) in \(n\) coordinates over a finite ring \(R\).
  278. Root lattice
    A lattice that is symmetric under a specific crystallographic reflection group; see [55; Table 4.1] for the list of crystallographic reflection groups and their corresponding root lattices. The root-lattice family consists of lattices \(A_n\), \(\mathbb{Z}^n\), or \(D_n\) for dimension \(n\), or \(E_{i}\) for \(i\in\{6,7,8\}\). Their generator matrices can be taken to be the root matrices of the corresponding reflection groups.
  279. Roth-Lempel code[389]
    Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.
  280. Row-Diagonal Parity (RDP) code[390]
    An MDS array code protecting against double erasures.
  281. Ruled-surface code[112,391]
    Evaluation code of polynomials evaluated on points lying on a ruled surface.
  282. Schubert code[392,393]
    Evaluation code of polynomials evaluated on points lying on a Schubert variety.
  283. Segre-variety RM-type code[394]
    Evaluation code of polynomials evaluated on points lying on a Segre variety.
  284. Self-dual additive code
    An additive \((n,2^n)_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product, usually the trace-Hermitian inner product.
  285. Self-dual code over \(R\)
    An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product.
  286. Self-dual linear code
    An \([n,n/2]_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to an inner product, most commonly either Euclidean or Hermitian. Self-dual codes exist only for even lengths and have dimension \(k=n/2\). A code that is equivalent to its dual is called iso-dual.
  287. Semakov-Zinoviev-Zaitsev (SZZ) equidistant code[395]
    Member of a family that is related to affine resolvable block designs and that is universally optimal.
  288. Sequential-recovery code[396,397]
    A \(t\)-erasure LRC whose coordinate erasures are recovered in sequential fashion.
  289. Sharp configuration[69,90,398] a.k.a. Delsarte code.
    A code \(C\) that attains a universal bound expressed in terms of the minimal distance, the number of distances between codewords, and the strength of the design formed by the codewords. For codes on a compact connected two-point homogeneous space, \(C\) is a design of strength \(M\) and admits \(m\) different distances between its points such that \(M \geq 2m - 1 - \delta\), where \(\delta\) is one if there are two antipodal points in \(C\) and zero otherwise [69].
  290. Simplex spherical code
    Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. Simplex spherical codewords in 2 (3, 4) dimensions form the vertices of a triangle (tetrahedron, 5-cell) In general, the code makes up the vertices of an \(n\)-simplex. See [60; Sec. 7.7] for a parameterization.
  291. Single parity-check (SPC) code a.k.a. Sum-zero code, Zero-sum code, Even-weight code.
    An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive. Its codewords are all even-weight binary strings. Its automorphism group is \(S_n\).
  292. Skew-cyclic code[399]
    A classical code \(C\) of length \(n\) over an alphabet \(R\) is skew-cyclic if there exists an automorphism, \(\theta\), of \(R\), such that for each string \(c_1 c_2 \cdots c_n\in C\), the skew-cyclically shifted string \(\theta(c_n) \theta(c_1) \cdots \theta(c_{n-1})\in C\). We say that \(C\) is a \(\theta\)-cyclic code over \(R\).
  293. Slepian group-orbit code[260,400,401]
    Spherical code in \(n\) dimensions whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the orthogonal group \(O(n)\) of rotations in \(n\) dimensions, i.e., the automorphism group of spherical codes under the Euclidean distance. Neither the vector nor the group are unique for a given code.
  294. Sloane-Whitehead code[402]
    Member of an infinite \((n,\lambda\cdot 2^{n-m-1},3)\) nonlinear code family for any \(n\) satisfying \(2^m \leq n < 3.2^{m-1}\) for some \(m\) and for \(\lambda\in\{20/16,19/16,18/16\}\). Such a code has more codewords than any linear code with the same length and distance. The code is constructed by applying the \((u|u+v)\) construction recursively to the Julin-Golay codes.
  295. Small-distance block code
    A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\).
  296. Smith \(40\)-point code[403,404]
    A \((10,40,1/6)\) spherical code found by W. D. Smith [403] and conjectured to be optimal in terms of minimizing potential energy functions [404]. A beautified set of coordinates can be found on the site [403].
  297. Snub-cube code
    Spherical \((3,24,0.55384)\) code whose codewords are the vertices of the snub cube.
  298. Spacetime block code (STBC)[290,405–407]
    In a space-time block code, \(n\) spatially separated channels transmit symbols to \(m\) receiving channel using \(T\) time slots. These symbols can be arranged in a \(n \times T\) matrix where the rows correspond to the channels, and the columns correspond to the time slots. The codewords \(\{X\}\) making up the code are thus \(n \times T\) matrices.
  299. Spacetime code (STC)[408]
    Code designed for wireless transmission of information (via, e.g., radio waves) such that the sender can send multiple times from multiple locations. A spacetime code uses a modulation scheme to encode a message into signals that are sent at different times through different antennas, thereby utilizing both spatial and temporal (i.e., spacetime) degrees of freedom.
  300. Spatially coupled LDPC (SC-LDPC) code[246–248,409,410]
    LDPC code whose parity-check matrix is constructed by "spatially" coupling several copies of a regular LDPC parity-check matrix in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. A finite-length chain is then capped by imposing either open boundary conditions (yielding non-tail-biting SC-LDPC codes) or open boundary conditions (yielding tail-biting SC-LDPC codes); sometimes extra terminating vertices are added to the ends of the chain. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant. These codes can be constructed, e.g., using the lifting procedure or using edge-cutting vectors [411].
  301. Sphere packing
    An analog code whose points can be thought of as forming centers of (real or complex) spheres that pack (real or complex) space. Sphere packings provide ways of encoding digital or analog information into the frequency, amplitude, and phase of one or more analog waveforms for transmission through, e.g., an optical fiber or free space. This is due to Kotelnikov's [412] and Shannon's [413] fundamental observation that a discretized electromagnetic signal of finite bandwidth and average power \(P\) can be represented as a vector in \(\mathbb{R}^n\) with norm \(nP\). Questions of capacity of electromagnetic communication channels then translate to packing problems in \(\mathbb{R}^n\) [55].
  302. Spherical code
    Code whose codewords are points on an \(n\)-dimensional sphere \(S^{n}\) with radius one.
  303. Spherical design[414]
    Spherical code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the real sphere. A spherical code is a spherical design of strength \(t\), i.e., a \(t\)-design, if the average of any polynomial of degree up to \(t\) over its codewords is equal to the average over the entire sphere.
  304. Spherical sharp configuration[69,90,415,416]
    A spherical code that is a spherical design of strength \(2m-1\) for some \(m\) and that has \(m\) distances between distinct points. All known spherical sharp configrations are either obtained from the Leech or \(E_8\) lattice, certain regular polytopes, or are CGS isotropic subspace spherical codes [417; Table 1].
  305. Square-antiprism code
    Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism.
  306. Srivastava code[49,164]
    A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\).
  307. Star code[418]
    An MDS array code protecting against triple erasures.
  308. Subspace code[419]
    A code that is a set of subspaces of \(GF(q)^n\).
  309. Subspace design[420,421] a.k.a. \(q\)-design, Geometric design.
    A \(q\)-ary code that can be mapped into a subspace \(t\)-\((n,w,\lambda)_q\) design.
  310. Sum-rank-metric code[422]
    A code whose performance is evaluated in the sum-rank metric, which is a metric that generalizes both the Hamming metric and the rank metric.
  311. Superimposed code[423–426]
    A set of binary strings such that taking a bitwise OR (e.g., \(1+1=1\)) of a small set of codewords does not yield another codeword.
  312. Suzuki-curve code[427]
    Evaluation AG code of rational functions evaluated on points lying on a Suzuki curve.
  313. Ta-Shma zigzag code[428]
    Member of a family of \(\epsilon\)-balanced codes that nearly achieves the asymptotic GV bound. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of order \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) [429].
  314. Tamo-Barg code[430]
    A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\).
  315. Tamo-Barg-Vladut code[431,432]
    Polynomial evaluation code on algebraic curves, such as Hermitian or Garcia-Stichtenoth curves, that is constructed to be an LRC. Codes can be constructed to be be able to recover locally after one or more erasures as well as to tackle the availability problem.
  316. Tanner code[385] a.k.a. Generalized LDPC (GLDPC) code.
    A linear \(q\)-ary code defined on a bipartite graph similar to a Tanner graph. This generalized Tanner graph consists of variable nodes and constraint nodes, with the generalization being that the constraint nodes of degree \(r\) now represent a linear codes of length \(r\).
  317. Tanner-Sridhara-Fuja (TSF) code[18]
    Array QC-LDPC code constructed from a cyclically shifted identity matrix; see [433; Exam. 21.6.5].
  318. Tensor-product code[85,434–436] a.k.a. Tensor code, Kroneckerian code, Product code.
    A matrix-based code constructed out of two linear binary or \(q\)-ary codes \(C_A,C_B\) in an outer-product construction denoted as \(C_A \otimes C_B\). Its dual is sometimes called the check-product code, denoted as \(C_{A}\boxplus C_{B}\).
  319. Ternary Golay code[170,437]
    A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [55] and sporadic simple groups [26]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters [171]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode.
  320. Tetracode[55]
    The \([4,2,3]_3\) self-dual MDS code that has connections to lattices [55].
  321. Tornado code[148,227,438]
    Linear binary code that is a precursor to fountain codes and whose encoding and decoding operations involve only XOR gates [439; Sec. 30.2].
  322. Torus-layer spherical code (TLSC)[440]
    Code whose codewords are elements of a foliation of the \(2n-1\)-dimensional hypersphere \(S^{2n-1}\) using flat tori \(S^1\times S^1\cdots\times S^1\). Related constructions include the spherical codes by Hopf foliations (SCHF) [441].
  323. Traceability code[442]
    An IPP code with which it is possible to detect a parent of a given pirated descendent by finding the closest codeword to that descendant.
  324. Tsfasman-Vladut-Zink (TVZ) code[443]
    Member of a family of residue AG or, more generally, evaluation AG codes where \(\cal X\) is either Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve.
  325. Turbo code[444,445]
    Code obtained from a parallel concatenation of two or more convolutional codes with permutations interleaving the individual encodings.
  326. Twisted BCH code[446–448] a.k.a. RS subspace subcode.
    Additive or linear \(q\)-ary code obtained from BCH codes via various lengthening and extension procedures such as Construction X and Construction XX.
  327. Two-weight code
    A linear \(q\)-ary code whose codewords all have one of two possible nonzero Hamming weights.
  328. Unary code a.k.a. Thermometer code.
    Trivial code that encodes integers \(1\) through \(n\) into binary strings of length \(n\). The \(i\)th codeword is a string consisting of \(i\) ones followed by \(n-i\) zeroes.
  329. Uniformly packed code[300,449,450]
    An \((n,K,2t+1)_q\) code is uniformly packed if its external distance is equal to \(t+1\) [26]; see [82; Def. 2.5] for the case of even distance and other generalizations.
  330. Unimodular lattice a.k.a. Self-dual lattice.
    A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\).
  331. Universally optimal \(q\)-ary code[90,398,451–455]
    A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on binary space [455].
  332. Universally optimal code[456]
    A code that produces a minimum over all codes of its cardinality for a large family of potential functions. Such codes exist for the conventional \(q\)-ary and real spaces (see children below), but can also be formulated for more exotic spaces such as Lie groups, projective spaces, and real Grassmanians [457,458].
  333. Universally optimal sphere packing[69]
    A periodic sphere packing that (weakly) minimizes all completely monotonic potentials of square Euclidean distance among all periodic packings of the same density.
  334. Universally optimal spherical code[69,454,459–461]
    A spherical code that (weakly) minimizes all completely monotonic potentials on the sphere for its cardinality. See [463][462; Sec. 12.4] for further discussion.
  335. Varshamov-Tenengolts (VT) code[464,465]
    Nearly optimal binary deletion-correcting code and code for the asymmetric channel.
  336. Vasilyev code[466]
    Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [26; pg. 77].
  337. Weight-two code[467]
    A length-\(n\) binary code whose codewords all have Hamming weight two. Such codes provide slightly extra redundancy for storage of small-scale information such as ZIP codes or decimal digits.
  338. Weighted-covering code
    A \(q\)-ary code for which balls of some radius centered at its codewords provide a weighted covering of the Hamming space.
  339. Witting polytope code a.k.a. \(4_{21}\) real polytope code.
    Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) real polytope, and the minimal lattice-shell code of the \(E_8\) lattice. The code is optimal and unique up to equivalence [55,468,469]. Antipodal pairs of points correspond to the 120 tritangent planes of a canonic sextic curve [69,202–204].
  340. Wozencraft ensemble code[470]
    A code that is part of the Wozencraft ensemble, a set of codes most of whose members achieve the GV bound.
  341. Wrapped spherical code[471]
    Spherical code in dimension \(n\) whose codewords are obtained from centers of spheres from a finite \(S^{n-1}\)-sphere packing of \(\mathbb{R}^{n}\) that is "wrapped" onto \(S^n\).
  342. X-code[472]
    An MDS array code with a simple geometrical construction that achieves optimal encoding and update complexity.
  343. Ye-Barg code[473,474]
    An MDS array code with the optimal access property; see Ref. [473] for definitions.
  344. Zetterberg code[475]
    Family of binary cyclic \([2^{2s}+1,2^{2s}-4s+1]\) codes with distance \(d>5\) generated by the minimal polynomial \(g_s(x)\) of \(\alpha\) over \(GF(2)\), where \(\alpha\) is a primitive \(n\)th root of unity in the field \(GF(2^{4s})\). They are quasi-perfect codes and are one of the best known families of double-error correcting binary linear codes
  345. Zigzag code[476]
    An MDS array code correcting against two erasures with optimal rebuilding ratio; see Ref. [476] for definitions.
  346. \((u|u+v)\)-construction code[402,477]
    Code constructed using a concatenation procedure that takes in two \(q\)-ary codes \(C_1,C_2\) and produces a new code whose codewords are \((u|u+v)\) for all \(u\in C_1\) and \(v\in C_2\). If the two codes have parameters \((n,K_1,d_1)\) and \((n,K_2,d_2)\), then the output code is a \((2n,K_1 K_2, \min\{2d_1,d_2\})\) code [32; Thm. 5.10][26; pg. 76].
  347. \(3_{21}\) polytope code[200] a.k.a. Hess polytope code, Hesse polytope code, 7-ic semi-regular figure code.
    Spherical \((7,56,1/3)\) code whose codewords are the vertices of the \(3_{21}\) real polytope (a.k.a. the Hess polytope). The vertices form the kissing configuration of the Witting polytope code. The code is optimal and unique up to equivalence [55,468,469]. Antipodal pairs of points correspond to the 28 bitangent lines of a general quartic plane curve [69,202–204].
  348. \(A_2\) triangular lattice a.k.a. \(A_2\) hexagonal lattice.
    Two-dimensional lattice that corresponds to the triangular tiling and that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. As a tiling, its dual (whose points lie at the centers of each triangle) is the honeycomb tiling.
  349. \(A_n\) lattice
    Lattice-based \(n\)-dimensional code that can be simply defined in \(n+1\) dimensions as the set of integer vectors \(x\) lying in the hyperplane \(x_0+x_1+\cdots+x_{n} = 0\).
  350. \(A_n^{\perp}\) lattice
    Lattice-based \(n\)-dimensional code whose codewords form the dual of the \(A_n\) lattice.
  351. \(BW_{32}\) Barnes-Wall lattice[43]
    BW lattice in dimension \(32\).
  352. \(BW_{32}\) lattice-shell code
    Spherical code whose codewords are points on the \(BW_{32}\) Barnes-Wall lattice normalized to lie on the unit sphere.
  353. \(D_3\) face-centered cubic (fcc) lattice a.k.a. Cannonball lattice.
    Laminated three-dimensional lattice consisting of layers of triangular lattices.
  354. \(D_4\) hyper-diamond lattice
    BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via Construction A.
  355. \(D_4\) lattice-shell code
    Spherical code whose codewords are points on the \(D_4\) lattice normalized to lie on the unit sphere.
  356. \(D_n\) checkerboard lattice
    Lattice consisting of all points whose coordinates add up to an even integer.
  357. \(ED_m\) code[478] a.k.a. Equidistant code with maximal distance.
    Member of the family of \( (c\frac{qt-1}{(t-1,q-1)},qt,ct\frac{q-1}{(t-1,q-1)}) \) \(q\)-ary codes, where \(c,t\geq 1\) and \((a,b)\) is the greatest common divisor of \(a\) and \(b\). Such codes are universally optimal and are related to resolvable block designs.
  358. \(E_6\) lattice-shell code
    Spherical code whose codewords are points on the \(E_6\) lattice normalized to lie on the unit sphere.
  359. \(E_6\) root lattice
    Lattice in dimension \(6\).
  360. \(E_7\) lattice-shell code
    Spherical code whose codewords are points on the \(E_7\) lattice normalized to lie on the unit sphere.
  361. \(E_7\) root lattice
    Lattice in dimension \(7\).
  362. \(E_8\) Gosset lattice[200]
    Unimodular even BW lattice in dimension \(8\), consisting of the Cayley integral octonions rescaled by \(\sqrt{2}\). The lattice corresponds to the \([8,4,4]\) Hamming code via Construction A.
  363. \(E_8\) Gosset lattice-shell code
    Spherical code whose codewords are points on the \(E_8\) Gosset lattice normalized to lie on the unit sphere.
  364. \(R\)-linear code
    A code of length \(n\) over a ring \(R\) is \(R\)-linear if it is a submodule of \(R^n\).
  365. \([2^m,m+1,2^{m-1}]\) First-order RM code a.k.a. Biorthogonal code, RM\((1,m)\) code, Augmented Hadamard code.
    A member of the family of first-order RM codes. Its codewords are the rows of the \(2^m\)-dimensional Hadamard matrix \(H\) and its negation \(-H\) with the mapping \(+1\to 0\) and \(-1\to 1\). They form a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping.
  366. \([2^m-1,m,2^{m-1}]\) simplex code[143,479] a.k.a. Shortened Hadamard code, RM\(^*(1,m)\) code, Maximum-length feedback-shift-register code.
    A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.
  367. \([2^r,2^r-r-1,4]\) Extended Hamming code[143,170,480]
    Member of an infinite family of binary linear codes with parameters \([2^r,2^r-r-1, 4]\) for \(r \geq 2\) that are extensions of the Hamming codes by a parity-check bit.
  368. \([2^r-1,2^r-r-1,3]\) Hamming code[170,480] a.k.a. RM\(^*(r-2,r)\) code.
    Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible non-zero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code.
  369. \([48,24,12]\) self-dual code
    An extended quadratic-residue code that is known to be the only self-dual doubly even code at its parameters [481].
  370. \([56,6,36]_3\) Hill-cap code[482]
    Projective two-weight ternary code based on the Games graph [484][483; Table 19.1] and Hill's 56-cap [482]. Its automorphism group contains \(PSL_3(4)\) [485].
  371. \([7,3,4]\) simplex code a.k.a. RM\(^*(1,3)\) code, Little Hamming code.
    Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.
  372. \([7,4,3]\) Hamming code[143,170,480]
    Second-smallest member of the Hamming code family.
  373. \([78,6,56]_4\) Hill-cap code[486]
    Projective two-weight quaternary code based on a cap that corresponds to a strongly regular graph [484; Table 7.1].
  374. \([8,4,4]\) extended Hamming code[143,170,480]
    Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code.
  375. \(\Lambda_{16}\) Barnes-Wall lattice[43]
    BW lattice in dimension \(16\).
  376. \(\Lambda_{16}\) lattice-shell code
    Spherical code whose codewords are points on the \(\Lambda_{16}\) Barnes-Wall lattice normalized to lie on the unit sphere.
  377. \(\Lambda_{24}\) Leech lattice[92]
    Even unimodular lattice in 24 dimensions that exhibits optimal packing. Its automorphism group is the Conway group \(.0\) a.k.a. Co\(_0\).
  378. \(\Lambda_{24}\) Leech lattice-shell code[92]
    Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code, and recursively taking their kissing configurations yields the \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes [414]; all codes are optimal and unique for their parameters [468,469].
  379. \(\mathbb{Z}^n\) hypercubic lattice
    Lattice-based code consisting of all integer vectors in \(n\) dimensions. Its generator matrix is the \(n\)-dimensional identity matrix. Its automorphism group consists of all coordinate permutations and sign changes.
  380. \(q\)-ary Hamming code[170,487]
    Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\).
  381. \(q\)-ary LDGM code
    \(q\)-ary linear code with a sparse generator matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) codes for which the number of nonzero entries in each row and column of the generator matrix are both bounded by a constant as \(n\to\infty\).
  382. \(q\)-ary LDPC code[488] a.k.a. Non-binary LDPC (NBDPC) code.
    A \(q\)-ary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
  383. \(q\)-ary code
    Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), i.e., \(q\)-ary strings. Error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric. The default distance is the Hamming distance \(d\), the weight (i.e., number of nonzero coordinates) of the lowest-weight nonzero codeword; such codes are usually denoted as \((n,K,d)_q\). The corresponding Hamming metric between two \(q\)-ary strings is the number of coordinates in which they differ. Unless stated otherwise, the distance for this class is the Hamming distance.
  384. \(q\)-ary code over \(\mathbb{Z}_q\)
    A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\).
  385. \(q\)-ary duadic code[62,489–491]
    Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist only when \(q\) is a square modulo \(n\) [62].
  386. \(q\)-ary linear LCC
    A linear code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)).
  387. \(q\)-ary linear LTC
    A \(q\)-ary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\). More technically, the code is a \((u,R)\)-LTC if the rows of its parity-check matrix \(H\in GF(q)^{r\times n}\) have weight at most \(u\) and if \begin{align} \frac{1}{r}|H x| \geq \frac{R}{n} D(x,C) \tag*{(10)}\end{align} holds for any \(q\)-ary string \(x\), where \(D(x,C)\) is the \(q\)-ary Hamming distance between \(x\) and the closest codeword to \(x\) [492; Def. 11]. A code satisfying the above constraint without the weight-\(u\) restriction is called an \(R\)-testable code [493].
  388. \(q\)-ary linear code over \(\mathbb{Z}_q\)
    A linear code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\).
  389. \(q\)-ary parity-check code a.k.a. Sum-zero code, Zero-sum code.
    An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero.
  390. \(q\)-ary protograph LDPC code[494–497]
    A \(q\)-ary LDPC code whose parity-check matrix is constructed using the lifting procedure applied to the incidence matrix of a sparse graph called, in this context, a protograph. An ability to assign non-binary edge weight called edge scaling can also be used in code construction.
  391. \(q\)-ary quadratic-residue (QR) code
    Member of a quadruple of cyclic \(q\)-ary codes of prime length \(n\) where \(q\) is prime and a quadratic residue modulo \(n\). The codes are constructed using quadratic residues and nonresidues of \(n\). Extensions to prime-power \(q\) are also known [498,499].
  392. \(q\)-ary repetition code
    An \([n,1,n]_q\) code encoding consisting of codewords \((j,j,\cdots,j)\) for \(j \in GF(q)\). The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information.
  393. \(q\)-ary sharp configuration[90,398,455]
    A \(q\)-ary code that admits \(m\) different distances between distinct codewords and forms a design of strength \(2m-1\) or greater.
  394. \(q\)-ary simplex code[143,479] a.k.a. \(q\)-ary maximum-length feedback-shift-register code.
    An \([n,m,q^{m-1}]_q\) projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element.
  395. \(t\)-design a.k.a. Quadrature, Cubature, Averaging set.
    A code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the code's underlying space \(X\). In that way, the codewords form an approximation of the space. A code is a design on \(X\) of strength \(t\), i.e., a \(t\)-design on \(X\), if the average of any polynomial of degree up to \(t\) over its codewords is equal to the uniform average over all of \(X\).
  396. \(t\)-erasure LRC a.k.a. Multiple-erasure LRC.
    A code which admits local recoverability against more than one coordinate erasure.

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