On the List Decodability of Self-orthogonal Rank Metric Codes

TL;DR

This paper investigates the list decodability of self-orthogonal rank metric codes, demonstrating that they can be efficiently list decoded up to certain radii with manageable list sizes, similar to random codes.

Contribution

It proves that self-orthogonal rank metric codes are highly list decodable with high probability, extending known results to this special class of codes.

Findings

High probability list decodability up to fractional radius  with small list size

List size depends on  and q, at most O_{ au, q}(1/)

Codes of rate up to Gilbert-Varshamov bound are ( n, exponential in 1/) list decodable

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.