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

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 $R=(1-\rho)(1-\frac{n}{m}\rho)-\varepsilon$ is shown to be (with high probability) list-decodable up to fractional radius $\rho\in(0,1)$ with lists of size at most $\frac{C_{p,q}}{\varepsilon}$, 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. A full version of this paper is accessible at https://arxiv.org/abs/1710.11516