Here is a list of \(q\)-ary linear codes over the Galois field \(\mathbb{F}_q\) for \(q > 2\). For \(q=2\), see Binary linear codes.
| Code | Description |
|---|---|
| Alternant code | Given a length-\(n\) GRS code \(C\) over \(\mathbb{F}_{q^m}\), an alternant code is the \(\mathbb{F}_q\)-subfield subcode of the dual of \(C\); see [1; Ch. 12]. Its parity-check matrix is an alternant matrix. |
| Barg-Tamo-Vladut code | Evaluation AG code on algebraic curves built from a Galois cover \(\phi:Y\to X\), where the recovery sets are fibres over rational points of \(X\) that split completely in the cover [2; Def. 15.9.10][2; Thm. 15.9.14]. The Barg-Tamo-Vladut construction generalizes the Tamo-Barg construction from \(PG(1,q)\) to longer AG codes, and variants can be built with higher local distance or availability \(2\) via fibre products of curves [2; Thm. 15.9.19][2; Thm. 15.9.21]. |
| Ben-Sasson-Sudan code | 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 \(c\). |
| Berlekamp code | A linear \(p\)-ary code (for prime \(p\)) 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 \(\mathbb{F}_p\) and then taking the subfield subcode of the corresponding code; see [3; Ch. 10.6]. |
| Bose–Chaudhuri–Hocquenghem (BCH) code | A cyclic \(q\)-ary code, with \(n\) and \(q\) relatively prime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). |
| Cartier code | A generalization of the Goppa codes to codes defined from curves of nonzero genus. Each code is a subcode of a certain residue AG code and is constructed using the Cartier operator. |
| Chien-Choy generalized BCH (GBCH) code | 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 \(r = \deg G \leq n-1\). |
| Classical fractal liquid code | 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 prime \(p\) and \(\epsilon > 0\) that is based on \(p\)-ary generalizations of the Sierpinski triangle. |
| Complete-intersection RM-type code | Evaluation code of polynomials evaluated on points lying on a complete intersection. |
| Cross-interleaved RS (CIRS) code | 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 [4]. |
| 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. |
| Deligne-Lusztig code | Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig variety, often a Deligne-Lusztig curve in the classical one-dimensional cases. |
| Denniston code | Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{2^a}\) codes for \(i < a\) constructed using Denniston arcs [5; Sec. 19.7.3]. |
| Divisible code | 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) [6,7]. 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. |
| Dual 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. |
| Elliptic code | Evaluation AG code of rational functions evaluated on points lying on an elliptic curve, i.e., a curve of genus one. |
| Evaluation AG code | Evaluation code over \(\mathbb{F}_q\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) lying on an algebraic curve \(\cal X\) defined over \(\mathbb{F}_q\), where the corresponding vector space \(L\) of functions \(f\) consists of certain rational functions (or, in special cases, polynomials). |
| Evaluation code | 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. |
| Extended GRS code | A GRS code extended by one extra coordinate to form an \([n+1,k,n-k+2]_q\) MDS code. In projective language, this corresponds to adding one more evaluation point, often interpreted as the point at infinity; in suitable equivalent descriptions, one may instead use an affine point such as \(0\). 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. |
| Flag-variety code | Evaluation code of polynomials evaluated on points lying on a flag variety. |
| Folded RS (FRS) code | A code obtained from an RS code by bundling consecutive symbols into larger alphabet symbols. This preserves the algebraic structure of the parent RS code while lowering the block length over an extension alphabet, and it is a key ingredient in capacity-approaching list-decoding constructions. |
| Generalized RM (GRM) code | Extensions of RM codes to \(q\)-ary coordinates that can be described as multivariate polynomial evaluation codes over affine or projective space. |
| 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 [2; Def. 15.3.19]. |
| Generalized Srivastava code | 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 \(\mathbb{F}_{q^m}\). |
| Goppa code | A linear \(q\)-ary code defined from a polynomial \(G(x)\) over an extension field and a set of evaluation points \(L\) avoiding the roots of \(G\). Goppa codes form a central family of alternant codes, admit efficient algebraic decoding algorithms, and include the binary Goppa codes used in the McEliece cryptosystem. When the base field equals the coefficient field, they coincide with residue AG codes on \(PG(1,q^m)\); in general, classical Goppa codes are subfield subcodes of such AG codes [2; Rem. 15.3.27][2; Thm. 15.3.28]. |
| Grassmannian evaluation code | Evaluation code of polynomials evaluated on points lying on a finite-field Grassmannian embedded into projective space using the Plucker embedding [8,9]. |
| Griesmer code | A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality. |
| Group-algebra code | An \( [n,k]_q \) code associated with a finite group \(G\) of order \(n\), viewed as an ideal in the group algebra \(\mathbb{F}_q[G]\) [10; Def. 16.4.3]. Equivalently, after identifying the \(n\) coordinate positions of each codeword with elements of \(G\), the code is invariant under the regular action of \(G\) and thus becomes a \(G\)-submodule of \(\mathbb{F}_q^n\) [12][11; Lemma 2.3]. A group-algebra code for an Abelian group is called an Abelian group-algebra code. |
| Hansen toric code | 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. |
| Hermitian code | Evaluation AG code of rational functions on a Hermitian curve over \(\mathbb{F}_{q^2}\). |
| Hermitian-hypersurface code | Evaluation code of polynomials evaluated on points lying on a Hermitian hypersurface. |
| Hill projective-cap code | Member of a projective code family that contains two \(q\)-ary sharp configurations [13; Table 12.1] and that is constructed using projective caps. |
| Hirschfeld code | A \([q+1,4,q-2]_q\) projective geometry code for non-prime \(q\) that is an example of an MDS code that is not an RS code; see [14; Exam. 7.6] for the generator matrix. |
| Hyperbolic evaluation code | An evaluation code over polynomials in two variables. Generator matrices are determined in Ref. [15] following initial formulations of the codes as generalized concatenations of RS codes [16,17]; see [18; Exam. 4.26]. |
| Hyperoval code | Projective code constructed from a hyperoval in the projective plane \(PG(2,q)\), where \(q\) is even. Since a hyperoval is a set of \(q+2\) points with no three collinear, the corresponding projective code has parameters \([q+2,3,q]_q\) [5; Exam. 19.2.1]; the \([6,3,4]_4\) hexacode is the smallest example. |
| Incidence-matrix projective code | A projective 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 [19,20][21; Sec. 14.4]. More generally, columns of a code’s parity-check matrix can also be organized as an incidence matrix. |
| Interleaved RS (IRS) code | A modification of RS codes where multiple polynomials are used to define each codeword. |
| Klein-quartic code | Evaluation AG code over \(\mathbb{F}_8\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) [18; Ex. 2.75]. |
| Linear \(q\)-ary code | An \((n,K,d)_q\) linear code is denoted as \([n,k,d]_q\), where \(k=\log_q K\) is 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 field elements \(\alpha,\beta \in \mathbb{F}_q\). This extra structure yields much information about their properties, making them a large and well-studied subset of codes. |
| Linear code with complementary dual (LCD) | A linear code \(C\) admits a complementary dual if \(C\) and its dual code \(C^{\perp}\) do not share any nonzero codewords. Equivalently, \(\mathbb{F}_q^n = C \oplus C^{\perp}\). |
| Maximum distance separable (MDS) code | A \(q\)-ary linear code whose parameters satisfy the Singleton bound with equality. |
| Meir code | 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. |
| Multiplicity code | A generalization of an \(m\)-variate polynomial evaluation code based on evaluating polynomials together with their Hasse derivatives up to order \(s-1\) at all points in \(\mathbb{F}_q^m\). Originally proposed for coding using the Rosenbloom-Tsfasman metric [22]. Univariate (\(m=1\)) [22,23] and multivariate (\(m>1\)) [24] codes have been proposed. |
| Narrow-sense RS code | 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 \(\mathbb{F}_q\). |
| Norm-trace code | 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. |
| Ovoid code | Member of a \([q^2+1,4,q^2-q]_q\) projective two-weight code family obtained from ovoids in \(\mathrm{PG}(3,q)\). If the columns of a generator matrix are the \(q^2+1\) points of an ovoid, then every hyperplane meets the ovoid in either \(1\) or \(q+1\) points, yielding the two nonzero weights \(q^2\) and \(q^2-q\). See [25; pg. 107][26; pg. 192] for further details. |
| Parity-check tensor-product code | A \(q\)-ary linear code constructed out of two \(q\)-ary linear codes with parity-check matrices \(H_A,H_B\) such that its parity-check matrix is \(H_A \otimes H_B\). Its dual has parity-check matrix \(G_A\otimes G_B\), where \(G_{A,B}\) are the generator matrices of the two underlying codes [27]. |
| Parvaresh-Vardy (PV) code | An IRS code with additional algebraic relations (a.k.a. correlations) between the codeword polynomials \(\{f^{(j)}\}_{j=1}^{t}\). These relations yield a list decoder that achieves list-decoding capacity. |
| Plane-curve evaluation code | Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve. |
| 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). |
| Primitive narrow-sense BCH code | A \(q\)-ary BCH code for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\). |
| Projective RM (PRM) code | Evaluation code obtained by evaluating homogeneous polynomials on the points of the projective space \(PG(m,q)\), equivalently on representatives of the nonzero vectors in \(\mathbb{F}_q^{m+1}\) whose leftmost nonzero coordinate is one. |
| Projective geometry code | Linear \(q\)-ary \([n,k,d]\) code whose 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 \(\mathbb{F}_q\). A choice of \(k\) linearly independent columns determines 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. |
| Projective two-weight code | A projective code whose codewords all have one of two possible nonzero Hamming weights. |
| Pyramid code | 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 [28; Sec. 31.3.1.1] for an example. |
| 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 \(n\) [29; Def. 3.2.8]. The codes are constructed using quadratic residues and nonresidues of \(n\). The definition extends to prime-power alphabet sizes and to prime-power lengths [30,31][29; Rem. 3.2.9]. A quadratic-residue code of prime length \(p\) has dimension \((p+1)/2\) [29; Sec. 3.2.1]. |
| Quadric code | Evaluation code of polynomials evaluated on points lying on a quadric hypersurface. |
| Quantum-inspired classical block code | A \(q\)-ary linear code whose construction was inspired by a quantum code. |
| Quasi group-algebra 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. |
| Reed-Solomon (RS) code | An \([n,k,n-k+1]_q\) linear code based on polynomials over \(\mathbb{F}_q\). |
| Residue AG code | Linear \(q\)-ary code defined using a set of \(\mathbb{F}_q\)-rational points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) on an algebraic curve \(\cal X\) and a linear space \(\Omega\) of certain rational differential forms \(\omega\) [2; Def. 15.3.2]. |
| Roth-Lempel code | Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes. |
| Ruled-surface code | Evaluation code obtained by evaluating global sections of a line bundle, or equivalently suitable polynomial functions, on rational points of a ruled surface over a finite field. Such codes extend algebraic-geometry constructions from curves to certain projective surfaces [32,33]. |
| Schubert evaluation code | Evaluation code of polynomials evaluated on points lying on a Schubert variety. |
| Segre-variety RM-type code | Evaluation code of multihomogeneous polynomials evaluated on points of a Segre variety, i.e., on the Segre embedding of a product of projective spaces. These codes are Reed-Muller-type analogues adapted to product projective geometries [34]. |
| 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 isodual. Any self-dual code is isodual, and hence formally self-dual [35; Rem. 4.2.2]. |
| Srivastava code | A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\). |
| Suzuki-curve code | Evaluation AG code of rational functions evaluated on points lying on a Suzuki curve. |
| Tamo-Barg code | A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\). |
| Tanner 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 \(r\). |
| Tsfasman-Vladut-Zink (TVZ) code | Member of a family of AG codes obtained from algebraic curves via the residue or evaluation construction. Sequences of curves with many rational points, such as Drinfeld modular curves, classical modular curves, or Garcia-Stichtenoth curves, yield the asymptotic parameters of the TVZ bound [2; Sec. 15.4.2]. |
| Two-weight code | A linear \(q\)-ary code whose codewords all have one of two possible nonzero Hamming weights [5; Def. 19.1]. |
| Wozencraft ensemble code | A code that is part of the Wozencraft ensemble, a set of codes most of whose members achieve the GV bound. |
| \([10,5,6]_9\) Glynn code | 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. |
| \([11,6,5]_3\) Ternary Golay code | A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [7] and sporadic simple groups [1]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code, whose weight enumerator is the Gleason polynomial \(g_5\) [35; Rem. 4.2.6]. Up to equivalence, both codes are unique for their respective parameters [36]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode [5; Exam. 19.3.2]. |
| \([2q+2,q+1]_3\) Pless symmetry 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. |
| \([4,2,3]_3\) Tetracode | The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [7]. Its weight enumerator is the Gleason polynomial \(g_4\) [35; Rem. 4.2.6]. |
| \([4,2,3]_4\) RS\(_4\) code | A Type II Euclidean self-dual extended RS code that is the smallest quaternary extended QR code [1; pg. 296][37; Sec. 2.4.2]. |
| \([5,3,3]_4\) Shortened hexacode | A perfect \([5,3,3]_4\) quaternary Hamming code that is the result of puncturing the hexacode [38]. |
| \([56,6,36]_3\) Hill-cap code | Projective two-weight ternary code based on the Games graph [39][5; Table 19.1] and Hill’s 56-cap [40]. Its automorphism group contains \(PSL(3,4)\) [41]. |
| \([6,3,4]_4\) Hexacode | The \([6,3,4]_4\) Hermitian self-dual MDS code that has connections to projective geometry, lattices [7], and conformal field theory [42]. Its weight enumerator is the Gleason polynomial \(g_7\) [35; Rem. 4.2.6]. |
| \([78,6,56]_4\) Hill-cap code | Projective two-weight quaternary code based on a cap that corresponds to a strongly regular graph [39; Table 7.1]. |
| \([n,n-1,2]_q\) \(q\)-ary parity-check 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. |
| \(q\)-ary Hamming code | 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 \(r \geq 2\) [29; (3.1)]. These are precisely the nontrivial perfect linear codes over \(\mathbb{F}_q\) [29; Thm. 3.3.1]. |
| \(q\)-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 \(n\to\infty\). |
| \(q\)-ary LDPC 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 \(n\to\infty\). |
| \(q\)-ary duadic code | Member of a pair of cyclic linear \(q\)-ary 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 \(n\) [43]. |
| \(q\)-ary linear LCC | A \(q\)-ary linear code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of a received word (assuming the corruption rate is within some tolerated threshold \(\delta\)). |
| \(q\)-ary linear LTC | A \(q\)-ary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\). |
| \(q\)-ary protograph LDPC code | 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. |
| \(q\)-ary repetition code | An \([n,1,n]_q\) code consisting of codewords \((j,j,\cdots,j)\) for \(j \in \mathbb{F}_q\). |
| \(q\)-ary simplex code | An \([n,m,q^{m-1}]_q\) equidistant 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. All nonzero simplex codewords have a constant weight of \(q^{m-1}\) [44,45]. |
References
- [1]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
- [2]
- A. Couvreur, H. Randriambololona, “Algebraic Geometry Codes and Some Applications.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [3]
- R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
- [4]
- M. Blaum, P. G. Farrell, and H. C. A. van Tilborg, “Array codes,” in Handbook of Coding Theory, Vol. II, Part 3, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 1855-1909
- [5]
- A. E. Brouwer, “Two-weight Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [6]
- S. Kurz, “Divisible Codes”, (2025) arXiv:2112.11763
- [7]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [8]
- D. Yu. Nogin, “Codes associated to Grassmannians”, Arithmetic, Geometry, and Coding Theory DOI
- [9]
- V. Lakshmibai and J. Brown, The Grassmannian Variety (Springer New York, 2015) DOI
- [10]
- W. Willems, “Codes in Group Algebras.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [11]
- A. Günther and G. Nebe, “Automorphisms of doubly even self-dual binary codes”, Bulletin of the London Mathematical Society 41, 769 (2009) arXiv:0810.3787 DOI
- [12]
- A. Günther. Automorphism groups of self-dual codes. PhD thesis, Aachen, Techn. Hochsch., 2009, 2009
- [13]
- P. Boyvalenkov, D. Danev, “Linear programming bounds.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [14]
- S. Ball, Finite Geometry and Combinatorial Applications (Cambridge University Press, 2015) DOI
- [15]
- O. Geil and T. Høholdt, “On Hyperbolic Codes”, Lecture Notes in Computer Science 159 (2001) DOI
- [16]
- K. Saints and C. Heegard, “On hyperbolic cascaded Reed-Solomon codes”, Lecture Notes in Computer Science 291 (1993) DOI
- [17]
- Gui-Liang Feng and T. R. N. Rao, “Improved geometric Goppa codes. I. Basic theory”, IEEE Transactions on Information Theory 41, 1678 (1995) DOI
- [18]
- T. Høholdt, J. H. van Lint, and R. Pellikaan, “Algebraic geometry codes,” in Handbook of Coding Theory, Vol. I, Part 1, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 871-961
- [19]
- B. Bagchi and S. P. Inamdar, “Projective Geometric Codes”, Journal of Combinatorial Theory, Series A 99, 128 (2002) DOI
- [20]
- M. Lavrauw, L. Storme, and G. Van de Voorde (2010), “Linear codes from projective spaces”. In A. Bruen & D. Wehlau (Eds.), Contemporary Mathematics (Vol. 523, pp. 185–202). Providence, RI, USA: American Mathematical Society (AMS)
- [21]
- L. Storme, “Coding Theory and Galois Geometries.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [22]
- M. Yu. Rosenbloom and M. A. Tsfasman, “Codes for the m-Metric”, Problemy Peredachi Informatsii, 33:1 (1997), 55–63; Problems of Information Transmission, 33:1 (1997), 45–52
- [23]
- R. R. Nielsen. List decoding of linear block codes. PhD thesis, Technical University of Denmark, 2001
- [24]
- S. Kopparty, S. Saraf, and S. Yekhanin, “High-rate codes with sublinear-time decoding”, Journal of the ACM 61, 1 (2014) DOI
- [25]
- R. Calderbank and W. M. Kantor, “The Geometry of Two-Weight Codes”, Bulletin of the London Mathematical Society 18, 97 (1986) DOI
- [26]
- J. Bierbrauer, Introduction to Coding Theory (Chapman and Hall/CRC, 2016) DOI
- [27]
- D. Ostrev, D. Orsucci, F. Lázaro, and B. Matuz, “Classical product code constructions for quantum Calderbank-Shor-Steane codes”, Quantum 8, 1420 (2024) arXiv:2209.13474 DOI
- [28]
- V. Ramkumar, M. Vajha, S. B. Balaji, M. N. Krishnan, B. Sasidharan, P. Vijay Kumar, “Codes for Distributed Storage.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [29]
- P. R. J. Östergård, “Construction and Classification of Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [30]
- J. van Lint and F. MacWilliams, “Generalized quadratic residue codes”, IEEE Transactions on Information Theory 24, 730 (1978) DOI
- [31]
- J. H. Lint, “Generalized quadratic-residue codes”, Algebraic Coding Theory and Applications 285 (1979) DOI
- [32]
- S. H. Hansen, “Error-Correcting Codes from Higher-Dimensional Varieties”, Finite Fields and Their Applications 7, 530 (2001) DOI
- [33]
- C. Lomont, “Error Correcting Codes on Algebraic Surfaces”, (2003) arXiv:math/0309123
- [34]
- M. González-Sarabia, C. Renterı́a, and H. Tapia-Recillas, “Reed-Muller-Type Codes Over the Segre Variety”, Finite Fields and Their Applications 8, 511 (2002) DOI
- [35]
- S. Bouyuklieva, “Self-dual codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [36]
- P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
- [37]
- Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [38]
- G. Höhn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [39]
- R. A. Games, “The packing problem for projective geometries over GF(3) with dimension greater than five”, Journal of Combinatorial Theory, Series A 35, 126 (1983) DOI
- [40]
- R. Hill (1973). “On the largest size of cap in s53”. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, 54(3), 378-384
- [41]
- N. Pace and A. Sonnino, “On linear codes admitting large automorphism groups”, Designs, Codes and Cryptography 83, 115 (2016) DOI
- [42]
- J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020) arXiv:2003.13700 DOI
- [43]
- V. Pless, “Duadic Codes and Generalizations”, Eurocode ’92 3 (1993) DOI
- [44]
- A. E.F. Jr. and H. F. Mattson, “Error-correcting codes: An axiomatic approach”, Information and Control 6, 315 (1963) DOI
- [45]
- E. Weiss, “Linear Codes of Constant Weight”, SIAM Journal on Applied Mathematics 14, 106 (1966) DOI