On the List Decodability of Self-orthogonal Rank Metric Codes

Abstract

V. Guruswami and N. Resch prove that the list decodability of ${\mathbb{F}}_q$-linear rank metric codes is as good as that of random rank metric codes in~\cite{venkat2017}. Due to the potential applications of self-orthogonal rank metric codes, we focus on list decoding of them. In this paper, we prove that with high probability, an $\F_q$-linear self-orthogonal rank metric code over ${\mathbb{F}}_q^{n\times m}$ of rate $R=(1-\tau )(1-\frac{n}{m}\tau )-\epsilon$ is shown to be list decodable up to fractional radius $\tau \in (0,1)$ and small $\epsilon \in (0,1)$ with list size depending on $\tau$ and $q$ at most $O_{\tau ,q}(\frac{1}{\epsilon })$. In addition, we show that an ${\mathbb{F}}_{q^m}$-linear self-orthogonal rank metric code of rate up to the Gilbert-Varshamov bound is $(\tau n,\exp (O_{\tau ,q}(\frac{1}{\epsilon })))$-list decodable.