Here is a list of binary linear codes.
| Code | Description |
|---|---|
| Accumulate-repeat-accumulate (ARA) code | 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. |
| Algebraic LDPC code | LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [1,2] or an algebraic structure such as a combinatorial design [3–5], balanced incomplete block design [6], a partial geometry [7], a generalized polygon [8,9], or a Latin square [10–12]. The extra structure and/or symmetry [13] of these codes can often be used to gain a better understanding of their properties. |
| Array-based LDPC (AB-LDPC) code | QC-LDPC code constructed deterministically from a disk array code known as a B-code. Its parity-check matrix admits a compact representation [14] and is related to RS codes. |
| Ben-Sasson-Goldreich-Harsha-Sudan-Vadhan (BGHSV) code | Locally testable \([n,k,d]\) code with \(n = k^{1+\epsilon}\) and query complexity of order \(O(1/\epsilon)\) for any \(\epsilon > 0\). |
| Ben-Sasson-Sudan-Vadhan-Wigderson (BSVW) code | 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 \(c\). |
| Binary BCH code | 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 \(r\geq 2\). |
| Binary duadic code | 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 \(8\) [15]. |
| Binary linear LTC | A binary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\). |
| 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 \(n\). |
| Block LDPC (B-LDPC) code | Member of a particular class of irregular QC-LDPC codes with efficient encoders. |
| Classical topological code | Classical code defined on a two-dimensional lattice and derived from a geometrically local stabilizer code, such as the surface code or color code. |
| Cycle LDPC code | An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns. |
| 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. |
| Cyclic redundancy check (CRC) code | 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. |
| Difference-set cyclic (DSC) code | Cyclic LDPC code constructed deterministically from a difference set. Certain DCS codes satisfy more redundant constraints than Gallager codes and thus can outperform them [16]. |
| Dinur code | 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 [17]. |
| Expander code | LDPC code whose parity-check matrix is derived from the adjacency matrix of bipartite expander graph [18] such as a Ramanujan graph or a Cayley graph of a projective special linear group over a finite field [19,20]. Expander codes admit efficient encoding and decoding algorithms and yield an explicit (i.e., non-random) asymptotically good LDPC code family. |
| Extended IRA (eIRA) code | A generalization of the IRA code in which the outer LDGM code is replaced by a random sparse matrix containing no weight-two columns. |
| Fibonacci code | 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. |
| Finite-geometry LDPC (FG-LDPC) code | 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 [21]. |
| Fountain code | 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. |
| Gallager (GL) code | 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. |
| Gauss' law code | 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 [22; 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 [22,23]. |
| Generalized Gallager code | 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. |
| Gold code | 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 \(r\) [24]. |
| Goldreich-Sudan code | 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 \(k^{1+o(1)}\). |
| Graph-adjacency code | Binary linear code whose generator matrix is that of the row space of 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. |
| Gray code | The first Gray code [25], 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 via what is known as the Gray map. |
| 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\(^*(1,m)\)). |
| Higman-Sims graph-adjacency code | A graph-based code whose generator matrix is constructed using the adjacency matrix of the Higman-Sims graph. This family includes a \([100,22,32]\) code whose dual is an optimal \([100,78,8]\) code [26; Table VI]. |
| Hoffman-Singleton graph-adjacency code | A graph-based code whose generator matrix is that of the row space of the adjacency matrix of the Hoffman-Singleton graph [27]. This \([50,22,7]\) code is dual to a \([50,28,8]\) code [26; Table III]. |
| Hsu-Anastasopoulos LDPC (HA-LDPC) code | 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. |
| Irregular LDPC code | An LDPC code whose parity-check matrix has a variable number of entries in each row or column. |
| Irregular repeat-accumulate (IRA) code | 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. |
| Justesen code | 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. |
| Karlin double circulant code | Member of the family of \([2m+2,m+1]\) double circulant codes such that \(m\) is prime of the form \(8k+3\) for some \(k\), and \(2m+2\) is a multiple of eight. See [28; Ch. 16] for their generator matrix. |
| Kasami code | Member of the family of \([2^{2r}-1, 3r, 2^{2r-1} - 2^{r-1} ]\) cyclic binary linear codes. |
| Kopparty-Meir-Ron-Zewi-Saraf (KMRS) code | 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 [29] showed that related concatenated codes achieve the GV bound. |
| Lazebnik-Ustimenko (LU) code | LDPC code whose Tanner graph comes from a particular family of \(q\)-regular graphs [30] of known girth and relatively large stopping sets. |
| Left-right Cayley complex code | Binary code constructed on a left-right Cayley complex using a pair of base codes \(C_A,C_B\) and an expander graph [18] 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. |
| 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. |
| Long code | 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 [31]. |
| 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. |
| Low-density parity-check (LDPC) 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 \(n\to\infty\). |
| Luby transform (LT) code | Erasure codes based on fountain codes. They improve on random linear fountain codes by having a much more efficient encoding and decoding algorithm. |
| MacKay-Neal LDPC (MN-LDPC) code | 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 \(g\) [32]. |
| Margulis LDPC code | Member of a class of LDPC codes deterministically constructed using an explicit class of regular graphs with no short cycles. Related explicit LDPC constructions [33] utilize Ramanujan graphs [19,20]. |
| Melas code | 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 [34]. |
| Multi-edge LDPC code | Irregular LDPC code whose construction generalizes those of the original examples of irregular LDPC as well as RA codes. |
| Newman-Moore code | 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 [35]. |
| One-hot 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. |
| Plaquette Ising code | Classical code defined on a two-dimensional square lattice whose parity checks are applied on the four vertices of each square. |
| Polar code | 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 \(x=u G^{\otimes n}\). |
| Protograph LDPC code | 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. |
| Quasi-cyclic LDPC (QC-LDPC) code | 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 [36]. Their simple structure makes them useful for several wireless communication standards. |
| Raptor (RAPid TORnado) code | 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. |
| Reed-Muller (RM) code | 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. |
| Regular LDPC code | An LDPC code whose parity-check matrix has a fixed number of entries for each row or column. |
| Regular binary Tanner code | A binary Tanner code defined on a regular bipartite graph, with the inner code being the same for all vertices. |
| Repeat-accumulate (RA) code | An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [37]. |
| Repeat-accumulate-accumulate (RAA) code | Generalization of the RA code in which two accumulators and permutations are used. |
| 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 \(S_n\). |
| Single parity-check (SPC) 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 \(S_n\). |
| Spatially coupled LDPC (SC-LDPC) code | 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 [38]. |
| Ta-Shma zigzag code | 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 \(n\to\infty\) [39]. |
| Tanner-Sridhara-Fuja (TSF) code | Array QC-LDPC code constructed from a cyclically shifted identity matrix; see [40; Exam. 21.6.5]. |
| Tornado code | Linear binary code that is a precursor to fountain codes and whose encoding and decoding operations involve only XOR gates [41; Sec. 30.2]. |
| Zetterberg code | 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 |
| \([23, 12, 7]\) Golay code | A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [42] and sporadic simple groups [28]. Up to equivalence, it is unique for its parameters [43]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [44]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [45,46]. |
| \([24, 12, 8]\) Extended Golay code | A self-dual \([24, 12, 8]\) code that is obtained from the Golay code by adding a parity check. Up to equivalence, it is unique for its parameters [43]. |
| \([2^m,m+1,2^{m-1}]\) First-order RM 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. |
| \([2^m-1,m,2^{m-1}]\) simplex 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. |
| \([2^r,2^r-r-1,4]\) Extended Hamming code | 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. |
| \([2^r-1,2^r-r-1,3]\) Hamming 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 nonzero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code. |
| \([48,24,12]\) self-dual code | An extended quadratic-residue code that is known to be the only self-dual doubly even code at its parameters [47]. |
| \([7,3,4]\) simplex 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. |
| \([7,4,3]\) Hamming code | Second-smallest member of the Hamming code family. |
| \([8,4,4]\) extended Hamming code | Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code. |
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