paper

Bounds on List Decoding of Rank-Metric Codes

arXiv:1301.4643 · doi:10.1109/TIT.2013.2274653

Abstract

So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length and minimum rank distance . The third bound proves that there exists a rank-metric code over $\Fqm$ of length such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on and similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.

10 pages, 2 figures, submitted to IEEE Transactions on Information Theory, short version presented at ISIT 2013

References in corpus (2)

Cited by in corpus (9)