On the List-Decodability of Random Linear Rank-Metric Codes

Abstract

The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an ${\mathbb{F}}_q$-linear rank-metric code over ${\mathbb{F}}_q^{m\times n}$ of rate $R=(1-\rho )(1-\frac{n}{m}\rho )-\epsilon$ is shown to be (with high probability) list-decodable up to fractional radius $\rho \in (0,1)$ with lists of size at most $\frac{C_{\rho ,q}}{\epsilon }$, where $C_{\rho ,q}$ is a constant depending only on $\rho$ and $q$. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, Håstad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting.