On the list decodability of rank-metric codes containing Gabidulin codes

TL;DR

This paper investigates the list decodability of rank-metric codes containing Gabidulin codes, establishing bounds and identifying families that cannot be efficiently decoded for large decoding radii.

Contribution

It provides bounds on list size for such codes and identifies specific families that are not efficiently list decodable, extending previous results.

Findings

Certain rank-metric codes containing generalized Gabidulin codes cannot be efficiently list decoded for large radii.

Identifies families of codes that are not efficiently list decodable.

Provides bounds on list size to determine the existence of polynomial-time list decoding algorithms.

Abstract

Wachter-Zeh in [42], and later together with Raviv [31], proved that Gabidulin codes cannot be efficiently list decoded for any radius $\tau$, providing that $\tau$ is large enough. Also, they proved that there are infinitely many choices of the parameters for which Gabidulin codes cannot be efficiently list decoded at all. Subsequently, in [41] these results have been extended to the family of generalized Gabidulin codes and to further family of MRD-codes. In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or not a polynomial-time list decoding algorithm exists. We detect several families of rank-metric codes containing a generalized Gabidulin code as subcode which cannot be efficiently list decoded for any radius large enough and families of rank-metric codes which cannot be efficiently list decoded. These results suggest that rank-metric codes which are $\mathbb{F}_{q^m}$-linear or that contains a (power of) generalized Gabidulin code cannot be efficiently list decoded for large values of the radius.