Higher-Order MDS Codes

.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

We consider (n, k , l) MDS codes of length n, dimension k, and subpacketization l over a finite field <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> . A codeword of such a code consists of n column-vectors of length l over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> , with the property that any k of them suffice to recover the entire codeword. Each of these n vectors may be stored on a separate node in a network. If one of the n nodes fails, we can recover its content by downloading symbols from the surviving nodes, and the total number of symbols downloaded in the worst case is called the repair bandwidth of the code. By the cut-set bound, the repair bandwidth of an (n, k , l) MDS code is at least l(n-1)/(n- k). There are several constructions of MDS codes whose repair bandwidth meets or asymptotically meets the cut-set bound. For example, letting r=n-k denote the number of parities, Ye and Barg constructed (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) Reed-Solomon codes that asymptotically meet the cut-set bound, Ye and Barg also constructed optimal-bandwidth and optimal-update (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) MDS codes. Wang, Tamo, and Bruck constructed optimal-bandwidth (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> n/(r+1)</sup> ) MDS codes, and these codes have the smallest known subpacketization for optimal-bandwidth MDS codes. A key idea in all these constructions is to represent certain integers in base r. We show how this technique can be refined to improve the subpacketization of the two MDS code constructions by Ye and Barg, while achieving asymptotically optimal repair bandwidth. Specifically, when r=s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> for an integer s, we obtain an (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+n-1</sup> ) Reed-Solomon code and an optimal-update (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS code, both having asymptotically optimal repair bandwidth. Thus for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , for example, we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> rather than 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i></sup> in the original constructions by Ye and Barg. When r is not an integral power, we can still obtain (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> ) Reed-Solomon codes and optimal-update ( n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS codes by choosing positive integers s and m such that s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m </sup> ≤ r. In this case, however, the resulting codes have bandwidth that is near-optimal rather than asymptotically optimal. We also present an extension of this idea to reduce the subpacketization of the Wang-Tamo-Bruck construction while achieving a repair-by-transfer scheme with asymptotically optimal repair bandwidth. For example, for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/r+ m-1</sup> , which significantly improves upon the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i>/(r+1)</sup> in the Wang-Tamo-Bruck construction. Based on the foregoing examples, we believe our approach may be generally useful in reducing the subpacketization of MDS code constructions that utilize r-ary expansion.

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.

A characterization of MDS codes that have an error correcting pair

MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>] MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at most <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>–1 zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.

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.

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of r+δ−1) are downloaded per server. The PIR scheme achieves the (download) rate R=(N−k−rt+1)/N, where N=gr is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that k+rt≤N. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

It is well known that a random q-ary code of rate Ω(e2) is list decodable up to radius (1 - 1/q - e) with list sizes on the order of 1/e2, with probability 1 - o(1). However, until recently, a similar statement about random linear codes has until remained elusive. In a recent paper, Cheraghchi, Guruswami, and Velingker show a connection between list decodability of random linear codes and the Restricted Isometry Property from compressed sensing, and use this connection to prove that a random linear code of rate Ω( e2 /log3(1/e)) achieves the list decoding properties above, with constant probability. We improve on their result to show that in fact we may take the rate to be Ω(e2), which is optimal, and further that the success probability is 1 - o(1), rather than constant. As an added benefit, our proof is relatively simple. Finally, we extend our methods to more general ensembles of linear codes. As an example, we show that randomly punctured Reed-Muller codes have the same list decoding properties as the original codes, even when the rate is improved to a constant.

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.

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.

A class of almost MDS codes

This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR) channels, erasure channels, AND (Z-) channels, OR channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L-1)-list decodable codes (where the list size L is a universal constant) exist for such channels. Our criterion essentially asserts that: For any given general adversarial channel, it is possible to construct positive rate (L-1)-list decodable codes if and only if the set of completely positive tensors of order-L with admissible marginals is not entirely contained in the order-L confusability set associated to the channel. The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1) extracting approximately equicoupled subcodes (generalization of equidistant codes) from any using hypergraph Ramsey’s theorem, and 2) significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which may be of independent interest. Other results include 1) List decoding capacity with asymptotically large L for general adversarial channels; 2) A tight list size bound for most constant composition codes (generalization of constant weight codes); 3) Rederivation and demystification of Blinovsky’s [Blinovsky, 1986] characterization of the list decoding Plotkin points (threshold at which large codes are impossible) for bit-flip channels; 4) Evaluation of general bounds [Wang et al., 2019] for unique decoding in the error correction code setting.

The weight spectra of MDS codes of length $ n $ and dimension $ k $ over the arbitrary alphabets are studied. For all $ q $-ary MDS codes of dimension $ k $ containing the zero codeword, it is shown that all $ k $ weights from $ n $ to $ n-k+1 $ are realized. The remaining case $ n=q+k-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.

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.

We present an explicit construction of linear-time encodable and decodable codes of rate r which can correct a fraction (1 —re)/2 of errors over an alphabet of constant size depending only on e, for every 0 r 0. The error-correction performance of these codes is optimal as seen by the Singleton bound (these are codes). Such near-MDS linear-time codes were known for the decoding from erasures [2]; our construction generalizes this to handle errors as well. Concatenating these codes with good, constant-sized binary codes gives a construction of linear-time binary codes which meet the so-called Zyablov bound. In a nutshell, our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding, but in addition have linear time encoding and decoding algorithms.We also obtain some results for list decoding targeted at the situation when the fraction of errors is very large, namely (1—e) for an arbitrarily small constant e > 0. The previously known constructions of such codes of good rate over constant-sized alphabets either used algebraic-geometric codes and thus suffered from complicated constructions and slow decoding, or as in the recent work of the authors [9], had fast encoding/decoding, but suffered from an alphabet size that was exponential in 1/e. We present two constructions of such codes with rate close to Ω(e2) over an alphabet of size quasi-polynomial in 1/e. One of the constructions, at the expense of a slight worsening of the rate, can achieve an alphabet size which is polynomial in 1/e. It also yields constructions of codes for list decoding from erasures which achieve new trade-offs. In particular, we construct codes of rate close to the optimal Ω(e) rate which can be efficiently list decoded from a fraction (1—e) of erasures.