Bounds on List Decoding of Rank-Metric Codes

TL;DR

This paper establishes bounds on the list size for rank-metric codes, showing that polynomial-time list decoding beyond certain radii is unlikely, highlighting fundamental differences from Hamming metric codes.

Contribution

It provides the first exponential bounds on list size for rank-metric codes, demonstrating the limitations of polynomial-time list decoding for Gabidulin codes.

Findings

No polynomial-time list decoding beyond Johnson radius for Gabidulin codes.

Exponential upper bound on list size for any rank-metric code.

Existence of rank-metric codes with exponential list size beyond half the minimum distance.

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 $n$ and minimum rank distance $d$. The third bound proves that there exists a rank-metric code over $\Fqm$ of length $n \leq m$ 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 $n$ and $d$ similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.