7 A Unified Framework for List Decoding of Algebraic Codes

We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC'13). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes.

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

This chapter, as well as the next one, explore the relation between the list decoding radius and minimum distance of a code. Understanding the relation between these parameters is useful for two reasons: (a) for several important families of codes like Reed-Solomon codes, we have precise bounds on the distance, and one can use the relation between list decoding radius and distance to understand the list decoding potential of these codes; and (b) this shows that one approach to construct good list decodable codes is to construct large distance codes, and the latter is a relatively well-studied and better understood problem. Also, historically the most significant algorithmic results on list decoding have been fueled by an attempt to decode codes whose good minimum distance highlighted their good combinatorial list decodability properties.

.We show new and improved list decoding properties of folded Reed–Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the source of recent advances in coding theory: folded RS codes were the first known explicit construction of capacity-achieving list decodable codes [V. Guruswami and A. Rudra, IEEE Trans. Inform. Theory, 54 (2008), pp. 135–150], and multiplicity codes were the first construction of high-rate locally decodable codes [S. Kopparty, S. Saraf, and S. Yekhanin, J. ACM, 61 (2014), 28]. In this work, we show that folded RS codes and multiplicity codes are in fact better than previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant [V. Guruswami and C. Wang, IEEE Trans. Inform. Theory, 59 (2013), pp. 3257–3268], [Z. Dvir and S. Lovett, Proc. 44th STOC, ACM, 2012, 351–358]. The list size we obtain is \((1/\varepsilon )^{O(1/\varepsilon )}\) where \(\varepsilon\) is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and we use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance. By known reductions, this gives, in turn, capacity-achieving locally list decodable codes with query complexity \(\exp ( \tilde O((\log N)^{5/6}))\) . This improves on the tensor-based construction of [B. Hemenway, N. Ron-Zewi, and M. Wootters, SIAM J. Comput., 49 (2019), pp. 157–195], which gave capacity-achieving locally list decodable codes of query complexity \(N^{\tilde O(1/\log \log N)}\) , and is close to the best known query complexity of \(\exp ( \tilde O(\sqrt{\log N}))\) for high-rate locally (uniquely) decodable codes [S. Kopparty et al., J. ACM, 64 (2017), 11].Keywordserror–correcting codeslist decodinglocal decodingfolded Reed–Solomon codesmultiplicity codesMSC codes11T7194B0594B3594B6094B65

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to compress, in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here l denotes the length of the list; assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

We show combinatorial limitations on efficient list decoding of Reed-Solomon codes beyond the Johnson-Guraswami-Sudan bounds. In particular, we show that for arbitrarily large fields FN, |FN| = N, for any ? ? (0,1), and K = N?: (1) Existence: there exists a received word wN : FN ? FN that agrees with a super-polynomial number of distinct degree K polynomials on ? N?? points each; (2) Explicit: there exists a polynomial time constructible received word w'N : FN ? FN that agrees with a superpolynomial number of distinct degree K polynomials, on ?2?(log N) K points each. In both cases, our results improve upon the previous state of the art, which was ? N?/? points of agreement for the existence case (proved by Justesen and Hoholdt), and ? 2N? points of agreement for the explicit case (proved by Guruswami and Rudra). Furthermore, for ? close to 1 our bound approaches the Guruswami-Sudan bound (which is ?(N K)) and implies limitations on extending their efficient Reed-Solomon list decoding algorithm to larger decoding radius. Our proof is based on some remarkable properties of sub-space polynomials. Using similar ideas, we then present a family of low rate codes that are efficiently list-decodable beyond the Johnson bound. This leads to an optimal list-decoding algorithm for the family of matrix-codes.

The Reed-Solomon (RS) codes, proposed in 1960 by Irving Reed and Gustav Solomon as a subset of error-correcting codes, have many current applications. The most significant of which are data recovery in storage systems, including hard drives, minidiscs, CDs, DVDs, Google's GFS, BigTable, and RAID 6, as well as in communication systems such as DSL, WiMAX, DVB, ATSC, and satellite communications. Additionally, RS codes are used as Bar codes in management and advertising systems, such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec Code. Nowadays, RS codes over Galois Fields GF(2m) with base 2 are commonly used in these applications, with the GF(28) field being the most widely used. This allows all 256 values of a byte to be represented as a polynomial with 8 binary coefficients over GF(28). Considering RS codes as cyclic codes in GF(2m) fields, as well as the validity of mathematical dependencies in arbitrary field GF(pm), is a motivation to verify and generalize the idea of generating RS codes in a field with base other prime than 2. As a result, the paper derives the specific features of the construction of Reed-Solomon codes by considering them as a family of codes over any field GF(pm) whose base is a prime p other than 2. The paper also discusses the unique properties of basic arithmetic operations in the arbitrary field GF(pm), which arise from the non-uniqueness of the inverse elements a and -a in a field with base other than 2.

Let $k,n,m \in {\mathbb Z} ^{+}$ be integers such that $k\leq n \leq m$ , let $\mathrm {G}_{n,k}\in {\mathbb F} _{q^{m}}^{n}$ be a Delsarte-Gabidulin code. Recently, Wachter-Zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius $\tau $ , providing $\tau $ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in ${\mathbb F}_{q^{m}}^{n}$ . Some years later, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$ . As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by Trombetti and Zhou. By exploiting arguments such as those used by Raviv and Wachter-Zeh, when $n\mid m$ , we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +1$ , where $d$ is its minimum distance. Nonetheless, in all the examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to $\left \lfloor{ \frac {d-1}2 }\right \rfloor +2$ , where $d$ is its minimum distance. In particular, this leads to show infinite families of Gabidulin codes, with underlying parameters not already covered by the result of Raviv and Wachter-Zeh, having this decodability defect. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove our main result, that is any rank-metric code of ${\mathbb F}_{q^{m}}^{n}$ of order $q^{kn}$ with $n$ dividing $m$ , such that $4n-3$ is a square in $\mathbb {Z}$ and containing $\mathrm {G}_{n,2}$ , is not efficiently list decodable at some values of the radius $\tau $ .

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.

A latest coding scheme named polarization‐adjusted convolutional (PAC) codes is shown to approach the dispersion bound for the code (128,64) under list decoding. However, to achieve the near‐bound performance, the list size of list decoding needs to be excessively large, which leads to insufferable time complexity. In this paper, to improve the speed of list decoding, fast list decoders for PAC codes are proposed. Four types of constituent nodes are defined and fast list decoding algorithms are provided for each of them. Simulation results present that fast list decoding with three types of constituent nodes can yield exactly the same error‐correction performance as list decoding, and reduce more than 50% time steps for the code (128,64). Moreover, fast list decoding with four types of constituent nodes can further reduce decoding complexity with negligible performance degradation.

We present an explicit and efficient algebraic construction of capacity-achieving list decodable codes with both constant alphabet and constant list sizes. More specifically, for any R ∈ (0,1) and є>0, we give an algebraic construction of an infinite family of error-correcting codes of rate R, over an alphabet of size (1/є)O(1/є2), that can be list decoded from a (1−R−є)-fraction of errors with list size at most exp(poly(1/є)). Moreover, the codes can be encoded in time poly(1/є, n), the output list is contained in a linear subspace of dimension at most poly(1/є), and a basis for this subspace can be found in time poly(1/є, n). Thus, both encoding and list decoding can be performed in fully polynomial-time poly(1/є, n), except for pruning the subspace and outputting the final list which takes time exp(poly(1/є)) · poly(n). In contrast, prior explicit and efficient constructions of capacity-achieving list decodable codes either required a much higher complexity in terms of 1/є (and were additionally much less structured), or had super-constant alphabet or list sizes. Our codes are quite natural and structured. Specifically, we use algebraic-geometric (AG) codes with evaluation points restricted to a subfield, and with the message space restricted to a (carefully chosen) linear subspace. Our main observation is that the output list of AG codes with subfield evaluation points is contained in an affine shift of the image of a block-triangular-Toeplitz (BTT) matrix, and that the list size can potentially be reduced to a constant by restricting the message space to a BTT evasive subspace, which is a large subspace that intersects the image of any BTT matrix in a constant number of points. We further show how to explicitly construct such BTT evasive subspaces, based on the explicit subspace designs of Guruswami and Kopparty (Combinatorica, 2016), and composition.

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1<L<=2. Based on previous work, we provide formulae which allow to optimally (in terms of residual codeword error probability) exploit the erasure option of decoders with arbitrary L, if the decoder can be used z>=1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders.

In this paper, we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code C refers to list decoding with radius L, where L is the minimum of the distances between the received word r and any codeword in C. We consider the problem of determining the value of L as well as determining all the codewords at distance L. Our approach involves a parametrization of interpolating polynomials of a minimal Gröbner basis G . We present two efficient ways to compute G. We also show that so-called re-encoding can be used to further reduce the complexity. We then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by Wu. Besides, we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity, which results in overall savings in both memory and computation.

We strengthen the notion of "double samplers", first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders. The ABNNR code construction [IEEE Trans. Inform. Theory, 38(2):509--516, 1992] achieves large distance by starting with a base code $C$ with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code $C$ but to any string. In this case the resulting code is approximate-list-decodable, i.e. the output list contains an approximation to the original input. Our list-decoding algorithm works as follows: it uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

A new family of modified polar codes and a new permutation selection scheme for belief propagation list (BPL) decoding are presented. We first propose a new code construction methodology to interpolate between Reed-Muller (RM) codes and polar codes. By taking advantage of an existing partial order on bit-channels whose corresponding indices share the same Hamming weight, we analyze the complexity of the new construction method. It is shown that we need to compute the reliability of roughly a fraction 1/log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> N of all the bit-channels contained in the subset. Then, we explore a special family of factor-graph layer permutations called stable permutations (SPs) that preserve a specified information set when the corresponding bit permutations are applied to message bit indices. Simulation results show that the error-rate performance of the new family of codes is better than that of 5G polar codes under successive cancellation list (SCL) decoding. In addition, the SP selection scheme stands out as a preferred one for BPL decoding in terms of error-rate performance, while the average number of iterations per belief propagation (BP) decoder on the permuted factor graph is close to that of the original BP decoder when an early stopping condition is applied.