On the List-Decodability of Random Linear Rank-Metric Codes

Abstract

The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an F q -linear rank-metric code over F m×n q of rate R = (1 − ρ)(1 − n m ρ) − ε is shown to be (with high probability) list-decodable up to fractional radius ρ ∈ (0, 1) with lists of size at most Cρ,q ε , where C ρ,q is a constant depending only on ρ and q. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, Håstad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting. IntroductionAt its core, coding theory studies how many elements of a (finite) vector space one can pack subject to the constraint that no two elements are too close. Typically, the notion of closeness is that of Hamming distance, that is, the distance between two vectors is the number of coordinates on which they differ. In a rank-metric code, introduced in [Del78], codewords are matrices over a finite field and the distance between codewords is the rank of their difference. A linear rank-metric code is a subspace of matrices (over the field to which the matrix entries belong) such that every non-zero matrix in the subspace has large rank.Rank-metric codes have found applications in magnetic recording [Rot91], public-key cryptography [GPT91, Loi10, Loi17], and space-time coding [LGB03, LK05]. There has been a resurgence of interest in this topic due to the utility of rank-metric codes and the closely related subspace codes for error-control in random network coding [KK08, SKK08]. Decoding algorithms for rank-metric codes also have connections to the popular topic of low-rank recovery, specifically in a formulation where the task is to recover a matrix H from few inner products H, M with measurement matrices M [FS12]. Finally, the study of rank-metric codes raises additional mathematical and algorithmic

On the list decodability of self-orthogonal rank-metric codes

“…There has been some interesting findings on the list decodability for random F q -linear rank metric codes [17], [5]. An interesting direction is to see whether these new results can be applied to improve results on specific F q -linear rank metric codes.…”

List decoding of rank-metric and cover-metric codes

“…Recently, V. Guruswami and N. Resch decreased the list size of F q -linear rank-metric codes. In [20], the list decodability of random F q -linear rank-metric codes is shown to match that of a general F q -linear rank-metric code.…”

“…This chapter is based on the work in [37]. In [20], V. Guruswami and N. Resch proved that a random F q -linear rank-metric code is list decodable with list decoding radius attaining the Gilbert-Varshamov bound. On the other hand, in Hamming metric, random linear selforthogonal codes can be list decoded to the Gilbert-Varshamov bound with polynomial list size [30].…”

“…Lemma 10. (see in [20]) For all integers n ≤ m, every τ ∈ (0, 1) and = O( √ nm), there exists a constant C τ,q > 1 such that if X 1 , • • • , X are selected independently and uniformly at random from B R (0, τ n), then we have…”

“…There have been some interesting findings on the list decodability for random F q -linear rank-metric codes [20] and [45]. An interesting direction is to see whether this new result can be applied to improve results on specific F q -linear rank-metric codes.…”