Here is a list of classical codes that have notable decoders.
Name Decoder(s)
Alternant code Variation of the Berlekamp-Welch algorithm [1].Euclidean algorithm; see [2; Ch. 12] for more details.Guruswami-Sudan list decoder [3,4].
B-code Efficient decoding algorithm against erasures [5].
Balanced code Efficient decoder [6–8].
Binary BCH code Peterson decoder with runtime of order \(O(n^3)\) [9,10] (see exposition in Ref. [11]).Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [12,13] and modification by Burton [14]; see also [15,16].Sugiyama et al. modification of the extended Euclidean algorithm [17,18].Guruswami-Sudan list decoder [3,4].
Binary code For few-bit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\) in Hamming distance. The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).
Binary quadratic-residue (QR) code Algebraic decoder [19].
Block code Decoding an error-correcting code is equivalent to finding the ground state of some statistical mechanical model [20].
Bose–Chaudhuri–Hocquenghem (BCH) code Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [12,13,21] and modification by Burton [14]; see also [15,16].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [9,22] (see exposition in Ref. [11]).Sugiyama et al. modification of the extended Euclidean algorithm [17,18].
Concatenated code Generalized minimum-distance decoder [23].
Convolutional code Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [24].BCJR decoder, also a trellis-based decoder [25].
Cyclic linear \(q\)-ary code Meggitt decoder [26].Information set decoding (ISD) [27], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Permutation decoding [28].
Cyclic linear binary code Meggitt decoder [26].Information set decoding (ISD) [27], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Permutation decoding [28].
Cyclic redundancy check (CRC) code GRAND [29].
Delsarte-Goethals (DG) code Since the equivalent \(\mathbb{Z_4}\) codes are extended cyclic codes, efficient encoding and decoding is possible [30].
Determinant code For exact repair, the interior points of the storage-bandwidth trade-off curve can be shown to be the convex hull of \(k\) corner points described by \((\alpha_m,\beta_m)= (\binom{k}{m},\binom{k-1}{m-1})\) for \(m\in\{1,2,\cdots,k\}\).
EVENODD code Efficient decoding algorithm against two erasures [31].
Error-correcting output code (ECOC) Standard Hamming-distance decoding [32].Inverse Hamming decoding [33].Euclidean-distance decoding or attenuated Euclidean decoding [34].Loss-based decoding [35].Probabilistic-based decoding [36].
Evaluation AG code Generalization of plane-curve decoder [37]. Another decoder [38] was later showed to be equivalent in Ref. [39]. Application of several algorithms in parallel can be used to decode up to half the minimum distance [40,41]. Computational procedure implementing these decoders is based on an extension of the Berlekamp-Massey algorithm by Sakata [42–44].Decoder based on majority voting of unknown syndromes [45] decodes up to half of the minimum distance [46].List decoders generalizing Sudan's RS decoder by Shokrollahi-Wasserman [47] and Guruswami-Sudan [4].
Expander code Decoding can be done in \(O(n)\) runtime using a greedy flip decoder [48] (see also [49]). The algorithm consists of flipping a bit of the received word if it will result in a greater number of satisfied parity checks. This is repeated until a codeword is reached.'Find erasures and Decode' a.k.a. Viderman's algorithm correcting order \(\Omega(n)\) errors in order \(O(n)\) time [50].
Fibonacci code An efficient algorithm base on minimum-weight perfect matching [51], which can correct high-weight errors that span rows and columns of the 2D lattice, with failure rate decaying super-exponentially with \(L\).
Finite-dimensional error-correcting code (ECC) Capacity-achieving Guessing Random Additive Noise Decoding (GRAND) [52] (see also [53]).
Folded RS (FRS) code Guruswami and Rudra [54,55] achieved list-decoding up to \(1-\frac{k}{n}-\epsilon\) fraction of errors using the Parvaresh-Vardy algorithm [56]; see Ref. [57] for a randomized construction.List-decoding works up to the Johnson bound using the Guruswami-Sudan algorithm [58].Folded RS codes, concatenated with suitable inner codes, can be efficiently list-decoded up to the Blokh-Zyablov bound [54,59].
Fountain code Invert the fragment generator matrix resulting from the continuous encoding process. If exactly \(K\) packets are received, then the probability of decoding correctly is \(0.289\). Extra packets increase this probability exponentially. The decoding runtime is dominated by the matrix inversion step, which takes order \(O(n^3)\) time.
Gabidulin code Fast decoder based on a transform-domain approach [60].Algebraic list decoder that decodes up to the Singleton bound [61].
Generalized RS (GRS) code The decoding process of GRS codes reduces to the solution of a polynomial congruence equation, usually referred to as the key equation. Decoding schemes are based on applications of the Euclid algorithm to solve the key equation.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [12,13,21].Guruswami-Sudan list decoder [3,4] and modification by Koetter-Vardy for soft-decision decoding [62].Hard-decision decoder for errors within the Singleton bound [63].
Golay code Majority decoding for the extended Golay code [64].Decoder for the extended Golay code using the hexacode [65].Both Golay codes have a trellis representation and can thus be decoded using trellis decoding [66,67].Bounded-distance decoder requiring at most 121 real operations [68].
Gold code General decoding is done by building a sparse parity check matrix, followed by applying an iterative message passing alogirithm. [69].
Goppa code Algebraic decoding algorithms [70]. If \( \text{deg} G(x) = 2t \) , then there exists a \(t\)-correcting algebraic decoding algorithm for \( \Gamma(L,G) \).Sugiyama et al. modification of the extended Euclidean algorithm [17,18].Binary Goppa codes can be decoded using an RS-based decoder [71].List decoder for binary Goppa codes [72].
Hergert code Since the equivalent \(\mathbb{Z_4}\) codes are extended cyclic codes, efficient encoding and decoding is possible. [30,73].
Hermitian code Unique decoding using syndromes and error locator ideals for polynomial evaluations. Note that Hermitian codes are linear codes so we can compute the syndrome of a received vector. Moreover, akin to the error-locator ideals found in decoding RS codes, for the multivariate case we must define an error locator ideal \(\Lambda \) such that the variety of this ideal over \(\mathbb{F}^{2}_q\) is exactly the set of errors. The Sakata algorithm uses these two ingredients to get a unique decoding procedure [42].
Hexacode Bounded-distance decoder requiring at most 34 real operations [68].
Interleaved RS (IRS) code Decoder that corrects up to \(1-\frac{2k+n}{3n}\) fraction of random errors [74].Decoder that corrects up to \(1-(\frac{k}{n})^{2/3}\) fraction of random errors [75].
Irregular repeat-accumulate (IRA) code Linear-time decoder [76].
Justesen code Generalized minimum distance decoding [77].
Kerdock code Soft decision decoding involves extending the Fast Hadamard Transform decoding algorithm for the first-order RM code to Kerdock code [30].Complexity of soft decision decoding algorithm: \(4^m\) multiplications and \(m4^m\) additions [30,78].
Lattice-based code Spherical decoder [79,80].
Linear STC Sphere decoder [81–83].
Linear \(q\)-ary code Maximum likelihood (ML) decoding. This algorithm decodes a received word to the most likely sent codeword based on the received word. ML decoding of reduced complexity is possible for virtually all \(q\)-ary linear codes [84].Optimal symbol-by-symbol decoding rule [85].Information set decoding (ISD) [86], a probabilistic decoding strategy that essentially tries to guess \(k\) correct positions in the received word, where \(k\) is the size of the code. Then, an error vector is constructed to map the received word onto the nearest codeword, assuming the \(k\) positions are error free. When the Hamming weight of the error vector is low enough, that codeword is assumed to be the intended transmission.Generalized minimum-distance decoder [23].Soft-decision maximum-likelihood trellis-based decoder [87].Random linear codes over large fields are list-recoverable and list-decodable up to near-optimal rates [88].Extensions of algebraic-geometry decoders to linear codes [89,90].
Linear binary code Decoding an arbitary linear binary code is \(NP\)-complete [91].Slepian's standard-array decoding [92].Recursive maximum likelihood decoding [93].Deep learning [94] and a transformer graph neural net (GNN) for soft decoding [95].Chase decoding, which uses channel measurement information [96].
Linear code with complementary dual (LCD) The decoding problem reduces to finding the nearest codeword in \(C\) given a word in \(C^{\perp}\) [97].
Linearized RS code Berlekamp-Welch-type decoder [98] and its sum-rank version [99].
Locally decodable code (LDC) LDCs admit local decoders, i.e., decoders whose runtime scales polylogarithmically with \(n\).
Low-density parity-check (LDPC) code Message-passing algorithm called belief propagation (BP) [100–102] (see also [103–105]).Soft-decision Sum-Product Algorithm (SPA) [100,103,106] and its simplification the Min-Sum Algorithm (MSA) [107].Linear programming [108–110].Iterative LDPC decoders can get stuck at stopping sets of their Tanner graphs [111], with decoder performance improving with the size of the smallest stopping set; see [112; Sec. 21.3.1] for more details. The smallest stopping set size can reach the minimum distance of the code [113].Ensembles of random LDPC codes under iterative decoders are subject to the concentration theorem [103,114]; see [112; Thm. 21.7.1] for the case of the BEC.Reinforcement learning [115].
Low-rank parity-check (LRPC) code Efficient probabilistic decoder [116].Mixed decoder [117].
Luby transform (LT) code Sum-Product Algorithm (SPA), often called a peeling decoder [118,119], similar to belief propagation [120].
MacKay-Neal LDPC (MN-LDPC) code Free-energy minimization and a BP decoder [121].
Matrix-product code Decoder up to half of the minimum distance for NSC codes [122].
Melas code Algebraic decoder [123].
Multiplicity code Multivariate multiplicity codes can be decoded up to half of the minimum distance in polynomial time [124,125].Univariate [126] and multivariate [124] multiplicity codes can be list-decoded up to the Johnson bound. Certain univariate code families achieve the list-decoding capacity for sufficiently large field characteristic [124,127].
Newman-Moore code Efficient decoder [128].
Orthogonal Spacetime Block Code (OSTBC) Maximum-likelihood decoding can be achieved with only linear processing [129].
Parvaresh-Vardy (PV) code PV codes can be list-decoded up to \(1-(t k/n)^{1/(t+1)}\) fraction of errors. This result improves over the Guruswami-Sudan algorithm for ordinary RS codes, which list-decodes up to \(1-\sqrt{k/n}\) fraction of errors.
Permutation spherical code Efficient maximum-likelihood decoder determining the Voronoi region of an error word.
Plane-curve code Generalization of the Peterson algorithm for BCH codes [37,130,131].
Polar code Successive cancellation (SC) decoder [132].Successive cancellation list (SCL) decoder [133] and a modification utilizing sequence repetition (SR-List) [134].Soft cancellation (SCAN) decoder [135,136].Belief propagation (BP) decoder [137].Noisy quantum gate-vased decoder [138].
Preparata code Preparata Codes can be decoded using a syndrome calculation based algorithm to correct all error patterns of Lee weight atmost 2 and detect all/ some error patterns of Lee weight 3/ 4 [30,78].
Random code Ball-collision decoding [139].Information set decoding (ISD) [140] and Finiasz and Sendrier (FS-ISD) decoding [141].
Rank-metric code Polynomial-reconstruction Berlekamp-Welch based decoder [142].Berlekamp-Massey based decoder [143].
Raptor (RAPid TORnado) code Raptor codes can be decoded using inactivation decoding [144], a combination of belief-propogation and Gaussian elimination decoding.
Reed-Muller (RM) code Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors [145] (see also Ch. 13 of Ref. [2]).Sequential code-reduction decoding [146].Matrix factorization can be used to decode an RM\((n,n-3)\) code [147]; see [148].
Reed-Solomon (RS) code Decoding general RS codes is \(NP\)-hard [149].Although using iFFT has its counterpart iNNT for finite fields, the decoding is usually standard polynomial interpolation in \(k=O(n\log^2 n)\). However, in erasure decoding, encoded values are only erased in \(r\) points, which is a specific case of polynomial interpolation and can be done in \(O(n\log n)\) by computing product of the received polynomial and an erasure locator polynomial and using long division to find an original polynomial. The long division step can be omitted to increase speed further by only dividing the derivative of the product polynomial, and derivative of erasure locator polynomial evaluated at erasure locations.Berlekamp-Massey decoder with runtime of order \(O(n^2)\) [12,13].Gorenstein-Peterson-Zierler decoder with runtime of order \(O(n^3)\) [9,22] (see exposition in Ref. [11]).Berlekamp-Welch decoder with runtime of order \(O(n^3)\) [150] (see exposition in Ref. [151]), assuming that \(t \geq (n+k)/2\).Gao decoder using extended Euclidean algorithm [152].Fast-Fourier-transform decoder with runtime of order \(O(n \text{polylog}n)\) [153].List decoders try to find a low-degree bivariate polynomial \(Q(x,y)\) such that evaluation of \(Q\) at \((\alpha_i,y_i)\) is zero. By choosing proper degrees, it can be shown such polynomial exists by drawing an analogy between evaluation of \(Q(\alpha_i,y_i)\) and solving a homogenous linear equation (interpolation). Once this is done, one lists roots of \(y\) that agree at \(\geq t\) points. The breakthrough Sudan list-decoding algorithm corrects up to \(1-\sqrt{2R}\) fraction of errors asymptotically in \(n\) [154]. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors [155]. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) [4], meaning that RS codes achieve list-decoding capacity; see Ref. [156] for bounds. It was later shown that generic RS codes achieve list-decoding capacity [157]. A modification of the Guruswami-Sudan algorithm by Koetter and Vardy is used for soft-decision decoding [62] (see also Ref. [158]). Subcodes of RS codes whose evaluation points lie in a subfield can be decoded up to the \(1-R\) [61]. List decoding of RS codes is known as noisy polynomial interpolation in cryptography [159].The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware [160] (see also Ref. [161], Ch. 5 and 10).
Regenerating code (RGC) If the recovered symbols are exactly equal to the erased symbols, the repair is called an exact repair.If the recovered symbols are not exactly equal to the erased symbols but still preserve the code properties, the repair is called a functional repair.
Regular binary Tanner code Parallel decoding algorithm corrects a fraction \(\delta_0^2/48\) of errors for Tanner codes [48]. A modification of said algorithm improves the fraction to \(\delta_0^2/4\) with no extra cost to complexity [162].Soft-decision linear-time decoder correcting errors almost up to half of the Blokh-Zyablov bound [163].
Repetition code Calculate the Hamming weight \(d_H\) of the code. If \(d_H\leq \frac{n-1}{2}\), decode the code as 0. If \(d_H\geq \frac{n+1}{2}\), decode the code as 1.Automaton-like decoders for the repetition code on a 2D lattice, otherwise known as the classical 2D Ising model, were developed by Toom [164,165]. An automaton by Gacs yields a decoder for a 1D lattice [166].
Single parity-check (SPC) code If the receiver finds that the parity information of a codeword disagrees with the parity bit, then the receiver will discard the information and request a resend.Wagner's rule yields a procedure that is linear in \(n\) [167] (see [168; Sec. 29.7.2] for a description).
Sphere packing Each signal point is assigned its own Voronoi cell, and a received point is mapped back to the center of the Voronoi cell in which it is located upon reception.
Subspace code List decoding up to the Singleton bound [61].
Ta-Shma zigzag code Unique and list decoders [169].
Tamo-Barg code Polynomial evaluation over \(r\) points [170].
Tanner code Min-sum and sum-product iterative decoders for binary Tanner codes [171,172]; see also [18,173]. These decoders can be improved using a probabilistic message-passing schedule [174].Any code can be put into normal form without significantly altering the underlying graph or the decoding complexity [175]. For an algebraic viewpoint on decoding, see [176].Iterative decoding is optimal for Tanner graphs that are free of cycles [172]. However, codes that admit cycle-free representations have bounds on their distances [177,178]; see [112,179].
Tensor-product code The simple decoding algorithm (first decode all columns with \(C_1\), then all rows with \(C_2\)) corrects up to \((d_A d_B-1)/4 \) errors.Algorithms such as generalized minimum-distance decoding [23] or the min-sum algorithm can decode all errors of weight up to \((d_A d_B-1)/2\). Error location may be coupled with Viterbi decoding for every faulty sub-block [180].
Ternary Golay code Decoder for the extended ternary Golay code using the tetracode [65].
Tornado code Linear-time peeling decoder [181]. This decoder either terminates when it has removed a given erasure pattern or when it is stuck in a stopping set.
Torus-layer spherical code (TLSC) Efficiently decodable [182].
Turbo code Turbo decoder [183], an instance of BP decoding [184].Maximum A Posteriori (MAP) decoder [185] and a soft output derivative [186]. The use of soft outputs can improve code performance [187].List decoding [188].VLSI intergrated-circuit decoding hardware [189].Autoencoder [190].
Varshamov-Tenengolts (VT) code Decoder based on checksums \(\sum_{i=1}^n i~x_i^{\prime}\) of corrupted codewords \(x_i^{\prime}\) [191,192].
Zetterberg code Kallquist first described an algebraic decoding theorem [193]. A faster version was later provided in Ref. [194] and further modified in Ref. [195].
\([2^m,m+1,2^{m-1}]\) First-order RM code First-order RM codes admit specialized decoders, such as the Fast Hadamard Transform decoder [196].
\([2^m-1,m,2^{m-1}]\) simplex code Serial orthogonal decoder [197,198]Quantum decoder [199].
\(q\)-ary code For small \(n\), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\). The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).
\(q\)-ary simplex code Permutation decoder [200] and MacDonald [201] codes.

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