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

Compared with classical block codes, list decoding rank-metric codes efficiently seems more difficult. The evidences to support this view include: 1) so far the only known efficient list decoding of rank-metric codes C gives decoding radius beyond (1 - R)/2 with positive rate R when the ratio of the number of rows over the number of columns is extremely small, 2) the Johnson bound for rank-metric codes does not exist as opposed to classical codes, and 3) the Gabidulin codes of square matrices cannot be list decoded beyond half of the minimum distance. Although the list decodability of random rank-metric codes and the limits to the list decodability have been determined completely, little work on efficient list decoding of rank-metric codes has been done. The only known efficient list decoding of rank-metric codes C gives decoding radius up to the singleton bound 1-R-e with positive rate R when ρ(C) is extremely small, i.e., O(ε
²
), where ρ(C) denotes the ratio of the number of rows over the number of columns of C. It is commonly believed that it is difficult to list decode rank-metric codes C with the ratio ρ(C) close to 1. The main purpose of this paper is to explicitly construct a class of rank-metric codes C with the ratio ρ(C) up to 1/2 and efficiently list decode these codes beyond unique decoding radius (1 - R)/2. Furthermore, encoding and list decoding algorithms run in polynomial time poly(n, exp(1/e)). The list size can be reduced to O(1/e) by randomizing the algorithm. Our key idea is to employ bivariate polynomials f (x, y), where f is linearized in variable y and the variable x is used to “fold” the code. In other words, the rows are used to correct rank errors and the columns are used to “fold” the code to enlarge the decoding radius. Apart from the above algebraic technique, we have to prune down the list. The algebraic idea enables us to pin down the messages into a structured subspace whose dimension is linear in the number n of columns. This “periodic” structure allows us to pre-encode the messages to prune down the list. More precisely, we use subspace design introduced by Guruswami and Xing to obtain a deterministic algorithm with a larger constant list size and employ hierarchical subspace-evasive sets introduced by Guruswami et al. to obtain a randomized algorithm with a smaller constant list size.