Subspace Polynomials and Limits to List Decoding of Reed–Solomon Codes

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.

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 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.

.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

Different variants of the code-based McEliece cryptosystem were proposed to reduce the size of the public key. All these variants use very structured codes, which open the door to new attacks exploiting the underlying structure. In this paper, we show that the dyadic variant can be designed to resist all known attacks. In light of a new study on list decoding algorithms for binary Goppa codes, we explain how to increase the security level for given public keysizes. Using the state-of-the-art list decoding algorithm instead of unique decoding, we exhibit a keysize gain of about 4 % for the standard McEliece cryptosystem and up to 21 % for the adjusted dyadic variant. 1

In the previous chapter we presented list decoding algorithms for two widely-studied families of algebraic codes: Reed-Solomon codes and AG-codes. Owing to the importance of these codes, these results can be viewed as providing strong evidence to the general utility of list decoding as an algorithmic notion. Indeed, as we shall see in future chapters, they set the stage for a whole body of results about list decoding.

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 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.

Locally decodable codes (LDCs) have played a central role in many recent results in theoretical computer science. The role of finite fields, and in particular, low-degree polynomials over finite fields, in the construction of these objects is well studied. However the role of group homomorphisms in the construction of such codes is not as widely studied. Here we initiate a systematic study of local decoding of codes based on group homomorphisms. We give an efficient list decoder for the class of homomorphisms from any abelian group G to a fixed abelian group H. The running time of this algorithm is bounded by a polynomial in log|G| and an agreement parameter, where the degree of the polynomial depends on H. Central to this algorithmic result is a combinatorial result bounding the number of homomorphisms that have large agreement with any function from G to H. Our results give a new generalization of the classical work of Goldreich and Levin, and give new abstractions of the list decoder of Sudan, Trevisan and Vadhan. As a by-product we also derive a simple(r) proof of the local testability (beyond the Blum-Luby-Rubinfeld bounds) of homomorphisms mapping \({\mathbb{Z}}_p^n\) to ℤ p , first shown by M. Kiwi.KeywordsAbelian GroupGroup HomomorphismOracle AccessDecodable CodeList DecoderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, a new list decoding algorithm for tail-biting convolutional codes (TBCCs) with a cyclic redundancy check (CRC) is proposed, where the CRC is considered as a concatenated outer code. The main idea of the proposed algorithm is to modify the list decoding procedure of the TBCC by using the CRC. Two algorithms are proposed for the list decoding of the TBCC with the CRC. The first proposed algorithm is a new initial state estimating algorithm using re-encoded CRC bits and having the low computational complexity. The other proposed algorithm is a modified list Viterbi algorithm, where trellis paths are fixed by re-encoded CRC bits and some CRC bits are used for the error correction. For the TBCC concatenated with the CRC code defined in the long-term evolution standard, the proposed decoding scheme by partially using CRC bits outperforms the conventional list decoding algorithms for the list size $L=4$ even though the proposed algorithm has the lower decoding complexity.

In Chapter 6, we introduced an efficient decoder for GRS codes, yet we assumed that the number of errors does not exceed ⌊( d −1)/2⌋, where d is the minimum distance of the code. In this chapter, we present a decoding algorithm for GRS codes, due to Guruswami and Sudan, where this upper limit is relaxed. When a decoder attempts to correct more than ⌊( d −1)/2⌋ errors, the decoding may sometimes not be unique; therefore, we consider here a more general model of decoding, allowing the decoder to return a list of codewords, rather than just one codeword. In this more general setting, a decoding is considered successful if the computed list of codewords contains the transmitted codeword. The (maximum) number of errors that a list decoder can successfully handle is called the decoding radius of the decoder. The approach that leads to the Guruswami–Sudan list decoder is quite different from the GRS decoder which was introduced in Chapter 6. Specifically, the first decoding step now computes from the received word a certain bivariate polynomial Q(x, z) over the ground field, F , of the code. Regarding Q(x, z) as a univariate polynomial in the indeterminate z over the ring F[x] , a second decoding step computes the roots of Q(x, z) in F[x] ; these roots are then mapped to codewords which, in turn, form the returned list.

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.

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.

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.

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.