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.