List-decoding of subspace codes and rank-metric codes up to Singleton bound

Abstract

Subspace codes and rank-metric codes can be used to correct errors and erasures in network, with linear network coding. Both types of codes have been extensively studied in the past five years.Subspace codes were introduced by Koetter and Kschischang to correct errors and erasures in networks where topology is unknown (the non-coherent case). In this model, the codewords are vector subspaces of a fixed ambient space; thus codes for this model are collections of such subspaces. In a previous work, we have developed a family of subspace codes, based upon the Koetter-Kschichang construction, which are efficiently list decodable. Using these codes, we achieved a better decoding radius than Koetter-Kschischang codes at low rates. Herein, we introduce a new family of subspace codes based upon a different approach which leads to a linear-algebraic list-decoding algorithm. The resulting error-correction radius can be expressed as follows: for any integer s, our list-decoder using s + 1-variate interpolation polynomials guarantees successful recovery of the message subspace provided the normalized dimension of errors is at most s(1 − sR). The same list-decoding algorithm can be used to correct erasures as well as errors. The size of output list is at most Q s−1 , where Q is the size of the field that message symbols are chosen from.Rank-metric codes are suitable for error correction in the case where the network topology and the underlying network code are known (the coherent case). Gabidulin codes are a well-known class of algebraic rank-metric codes that meet the Singleton bound on the minimum rank-distance of a code. In this paper, we introduce a folded version of Gabidulin codes analogous to the folded Reed-Solomon codes of Guruswami and Rudra along with a list-decoding algorithm for such codes. Our list-decoding algorithm makes it possible to recover the message provided that the normalized rank of error is at most 1 − R − , for any > 0. Notably this achieves the information theoretic bound on the decoding radius of a rank-metric code.

A New Class of Rank-Metric Codes and Their List Decoding Beyond the Unique Decoding Radius

“…This implies that decoding radius of list decoding the square Gabidulin codes is not better than unique decoding. Inspired by good list decodability of the folded Reed-Solomon codes [12], people started to consider list decoding of folded Gabidulin codes [19]. However, the rate of the folded Gabidulin code in [19] tends to 0.…”

“…Inspired by good list decodability of the folded Reed-Solomon codes [12], people started to consider list decoding of folded Gabidulin codes [19]. However, the rate of the folded Gabidulin code in [19] tends to 0. In 2013, Guruswami and Xing [17] considered subcodes of the Gabidulin codes via point evaluation in a subfield and showed that list decodability of subcodes of the Gabidulin codes achieves the Singleton bound τ = 1 − R. However, the ratio ρ = n/t of the rank-metric code C ⊆ M n×t (F q ) constructed by Guruswami and Xing [17] is Θ(ε 2 ).…”

“…However, if applying Sudan's list decoding idea to decoding of the Gabidulin codes, we get only unique decoding (see [18]). In order to enlarge list decoding radius of the Gabidulin codes, Mahdavifar and Vardy [19] considered folded Gabidulin codes. As a result, the rate tends to 0.…”

Row reduction applied to decoding of rank-metric and subspace codes

“…Mahdavifar-Vardy (MV) codes [25,27] are subspace codes constructed by evaluating powers of skew polynomials at certain points. We will describe how one can use row reduction to carry out the most computationally intensive step of the MV decoding algorithm given in [27], the Interpolation step.…”

“…[6]; to the best of our knowledge, analogous generalisations over skew polynomial rings have yet to see any applications.Lately, there has been an interest in Gabidulin codes over number fields, with applications to space-time codes and low-rank matrix recovery [3]. Their decoding can also be reduced to a shift-register type problem [29], which could be solved using the algorithms in this paper (though again, one should analyse the bit-complexity).Mahdavifar-Vardy (MV) codes [25,27] are subspace codes whose main interest lie in their property of being list-decodable beyond half the minimum distance. Their rate unfortunately tend to zero for increasing code lengths.…”