Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets

.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 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 Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code dimension. By pre-coding the message polynomials into a subspace-evasive set, we get a Monte Carlo construction of a subcode of Reed-Solomon codes that can be list decoded from a fraction (1-R-e) of errors in polynomial time (for any fixed e > 0) with a list size of O(1/e). Our methods extend to algebraic-geometric (AG) codes, leading to a similar claim over constant-sized alphabets. This matches parameters of recent results based on folded variants of RS and AG codes. but our construction here gives subcodes of Reed-Solomon and AG codes themselves (albeit with restrictions on the evaluation points).Further, the underlying algebraic idea also extends nicely to Gabidulin's construction of rank-metric codes based on linearized polynomials. This gives the first construction of positive rate rank-metric codes list decodable beyond half the distance, and in fact gives codes of rate R list decodable up to the optimal (1-R-e) fraction of rank errors. A similar claim holds for the closely related subspace codes studied by Koetter and Kschischang.We introduce a new notion called subspace designs as another way to pre-code messages and prune the subspace of candidate solutions. Using these, we also get a deterministic construction of a polynomial time list decodable subcode of RS codes. By using a cascade of several subspace designs, we extend our approach to AG codes, which gives the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-e fraction of errors over an alphabet of constant size (that depends only on e). The list size bound is almost a constant (governed by log* (block length)), and the code can be constructed in quasi-polynomial time.

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.

We consider transmissions of secure messages over a burst erasure wiretap channel under decoding delay constraint. For block codes we introduce and study delay optimal secure burst erasure correcting (DO-SBE) codes that provide perfect security and recover a burst of erasures of a limited length with minimum possible delay. Our explicit constructions of DO-SBE block codes achieve maximum secrecy rate. We also consider a model of a burst erasure wiretap channel for the streaming setup, where in any sliding window of a given size, in a stream of encoded source packets, the eavesdropper is able to observe packets in an interval of a given size. For that model we obtain an information theoretic upper bound on the secrecy rate for delay optimal streaming codes. We show that our block codes can be used for construction of delay optimal burst erasure correcting streaming codes which provide perfect security and meet the upper bound for a certain class of code parameters.

Recently, a construction of polynomial-time decodable codes for discrete memoryless multiple access channels (MACs) has been proposed based on multi-level concatenated coding. This paper provides several refinements to concatenated codes for MACs. First, an explicit construction of inner codes is devised based on an ensemble of affine codes so that no search for good inner codes that achieve the capacity region is needed. Then a new decoding algorithm is proposed in which the inner decoding and the outer decoding alternate to improve a lower bound on the error exponent. It is shown that the proposed codes still arbitrarily approach every interior point of the capacity region for a given MAC with polynomial-time decoding complexity.

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 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.

Non-malleable coding, introduced by Dziembowski et al. (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable coding is possible against any class of adversaries of bounded size. In particular, Dziembowski et al. show that such codes exist and may achieve positive rates for any class of tampering functions of size at most $$2^{2^{\alpha n}}$$22źn, for any constant $$\alpha \in [0, 1)$$źź[0,1). However, this result is existential and has thus attracted a great deal of subsequent research on explicit constructions of non-malleable codes against natural classes of adversaries. In this work, we consider constructions of coding schemes against two well-studied classes of tampering functions; namely, bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and the much more general class of split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). We obtain the following results for these models. (1) For bit-tampering adversaries, we obtain explicit and efficiently encodable and decodable non-malleable codes of length n achieving rate $$1-o(1)$$1-o(1) and error (also known as exact security) $$\exp (-\tilde{\varOmega }(n^{1/7}))$$exp(-Ω~(n1/7)). Alternatively, it is possible to improve the error to $$\exp (-\tilde{\varOmega }(n))$$exp(-Ω~(n)) at the cost of making the construction Monte Carlo with success probability $$1-\exp (-\varOmega (n))$$1-exp(-Ω(n)) (while still allowing a compact description of the code). Previously, the best known construction of bit-tampering coding schemes was due to Dziembowski et al. (ICS 2010), which is a Monte Carlo construction achieving rate close to .1887. (2) We initiate the study of seedless non-malleable extractors as a natural variation of the notion of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove a general result on existence of seedless non-malleable extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1 / 5 and exponentially small error. In a separate recent work, the authors show that the optimal rate in this model is 1 / 2. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate.

We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the shortening method and generalized Hamming weights that are fundamental parameters of any linear codes with many useful applications. This bound improves existing upper bounds. To reduce the gap in between upper and lower bounds, we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes. The first construction relies on expander graphs and is better in low rate region. The second construction utilizes the LRC codes developed by Wang et al. as inner codes and is better in high rate region. We also suggest one possible generalization of LRC codes in which the recovering sets can intersect in a small number of coordinates. This feature allows us to increase the achievable code rate and still meet load balancing requirements. We derive upper and lower bounds on the parameters of such codes and present explicit constructions of codes with such a property.

We propose an explicit construction of full-diversity space-time block codes, under the constraint of an error correction capability. Furthermore, these codes are constructed in order to be suitable for a serial concatenation with an outer linear forward error correcting (FEC) code. We apply the binary rank criterion, and we use the threaded layering technique and an inner linear FEC code to define a space-time error-correcting code. When serially concatenated with an outer linear FEC code, a product code can be built at the receiver, and adapted iterative receiver structures can be applied. An optimized hybrid structure mixing MMSE turbo equalization and turbo product code decoding is proposed. It yields reduced complexity and enhanced performance compared to previous existing structures.

In a DNA sequence, we have the celebrated Watson-Crick complement $\overline{T}=A, \overline{A}= {T}, \overline{C}= G$, and $\overline{G}= C$. The phenomenon of secondary structure refers to the tendency of a single stranded DNA sequence to fold back upon itself, and it is usually caused by the existence of two non-overlapping reverse complement consecutive subsequences, defined as follows. Given an integer $m \geq 2$, two non-overlapping consecutive subsequences of length m, denoted as $x=(x_{1}, x_{m})$ and $y=(y_{1}, y_{m})$, are called reverse complement if $x_{i}=\overline{y_{m-i+1}}$ for 1$\leq i \leq m$. The property of secondary structure avoidance (SSA) forbids a sequence to contain such reverse complement subsequences, and it is a key criterion in the design of single-stranded DNA sequences for DNA computing and storage. In this paper, we improve on a recent result of Nguyen et al., by introducing explicit constructions of secondary structure avoidance codes and analyzing the capacity for any given m. In particular, our constructions have optimal rate 1.1679bits/nt and 1.5515bits/nt when m = 2 and m + 3, respectively.

One of the directions to improve the efficiency of modern telecommunication systems is the transition to the use of multidimensional signals for continuous channels of information transmission. As a result of studies carried out in recent years, it has been established that it is possible to ensure high quality of information transmission in continuous channels by combining demodulation and decoding operations into a single procedure that involves the construction of a code construct for a multidimensional signal. This paper considers issues related to estimating the possibility to improve the efficiency of continuous information transmission channel by changing the signal distance of the code structure. It has been established that the code structures of such types as a hierarchical code construct of signals, a hierarchical code construct of signals with Euclidean metric, a reversible code construct of signals, a reversible code construct of signals with Euclidean metric have the potential, when used, to increase the speed of information transmission along a continuous channel. With a signal distance reduced by 10 percent or larger, it could increase by two times or faster. The estimation of the effect of reducing a signal distance on the efficiency of certain types of code structures was carried out. It has been established that the hierarchical reversible code construct, compared to the hierarchical code construct, provides a win of up to two or more times in the speed of information transmission with a halved signal distance. Implementing the modulation procedure has no fundamental difficulties, on the condition that for each code of the code construct the encoding procedure is known when using binary codes. The results reported here make it possible to build an acceptably complex demodulation procedure according to the specified types of code structures

In this paper, we extend the notion of locally recoverable codes with availability to secret sharing schemes. The main problem that we considered is how to store information using locally recoverable codes with all symbol locality and availability in such way that useful information can be recovered using an only small subset of coordinates while a user who observes less than a certain number of coordinates does not get any information. In other words, we have to protect locally recoverable codes with availability over passive eavesdropper that can observe only limited number of coordinates. Upper bounds on number of bits that can be securely stored in such systems together with explicit constructions of codes with such a property are proposed.

In this paper, we investigate a special tower of function fields, which is asymptotically bad. The primary goal is to construct explicitly a class of algebraic-geometry (AG) codes based on this tower. The so called Hermitian codes are codes corresponding to the 2nd function field in this tower.