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

TL;DR

This paper demonstrates that random linear rank-metric codes have list-decodability properties comparable to random rank-metric codes, with high probability, for rates close to the theoretical limit, using an adaptation of existing proof techniques.

Contribution

It extends the understanding of list-decodability to linear rank-metric codes, matching the bounds known for non-linear codes, and adapts proof methods from Hamming to rank-metric.

Findings

Linear rank-metric codes are list-decodable up to the same radius as random codes.

List sizes are bounded by a constant over epsilon, matching non-linear code bounds.

The proof adapts techniques from Hamming metric to rank-metric setting.

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 $R = (1-\rho)(1-\frac{n}{m}\rho)-\varepsilon$ is shown to be (with high probability) list-decodable up to fractional radius $\rho \in (0,1)$ with lists of size at most $\frac{C_{\rho,q}}{\varepsilon}$, 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, H\aa stad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting.