# On the list decodability of Rank Metric codes

## TL;DR

This paper establishes lower bounds on list decodability for various classes of rank metric codes, including Gabidulin and twisted Gabidulin codes, showing they cannot be efficiently list decoded beyond certain radii.

## Contribution

It extends previous bounds to a broader class of rank metric codes and identifies infinite families that are not efficiently list decodable at specific radii.

## Key findings

- Lower bounds for list decodability of generalized Gabidulin codes.
- Identification of infinite families of codes not efficiently list decodable.
- Application of subspace trinomials to prove non-decodability at certain radii.

## Abstract

Let $k,n,m \in \mathbb{Z}^+$ integers such that $k\leq n \leq m$, let $\mathrm{G}_{n,k}\in \mathbb{F}_{q^m}^n$ be a Delsarte-Gabidulin code. Wachter-Zeh proven that codes belonging to this family cannot be efficiently list decoded for any radius $\tau$, providing $\tau$ is large enough. This achievement essentially relies on proving a lower bound for the list size of some specific words in $\mathbb{F}_{q^m}^n \setminus \mathrm{G}_{n,k}$. In 2016, Raviv and Wachter-Zeh improved this bound in a special case, i.e. when $n\mid m$. As a consequence, they were able to detect infinite families of Delsarte-Gabidulin codes that cannot be efficiently list decoded at all. In this article we determine similar lower bounds for Maximum Rank Distance codes belonging to a wider class of examples, containing Generalized Gabidulin codes, Generalized Twisted Gabidulin codes, and examples recently described by the first author and Yue Zhou. By exploiting arguments suchlike those used in the above mentioned papers, when $n\mid m$, we also show infinite families of generalized Gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to $\left\lfloor \frac{d-1}2 \right\rfloor+1$, where $d$ is its minimum distance. Nonetheless, in all other examples belonging to above mentioned class, we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to $\left\lfloor \frac{d-1}2 \right\rfloor+2$, where $d$ is its minimum distance. Finally, relying on the properties of a set of subspace trinomials recently presented by McGuire and Mueller, we are able to prove that any rank metric code of $\mathbb{F}_{q^m}^n$ of order $q^{kn}$ with $n$ dividing $m$, such that $4n-3$ is a square in $\mathbb{Z}$ and containing $\mathrm{G}_{n,2}$, is not efficiently list decodable at some values of the radius $\tau$.