On List-decodability of Random Rank Metric Codes

Abstract

In the present paper, we consider list decoding for both random rank metric codes and random linear rank metric codes. Firstly, we show that, for arbitrary $0<R<1$ and $\epsilon>0$ ($\epsilon$ and $R$ are independent), if $0<\frac{n}{m}\leq \epsilon$, then with high probability a random rank metric code in $F_{q}^{m\times n}$ of rate $R$ can be list-decoded up to a fraction $(1-R-\epsilon)$ of rank errors with constant list size $L$ satisfying $L\leq O(1/\epsilon)$. Moreover, if $\frac{n}{m}\geq\Theta_R(\epsilon)$, any rank metric code in $F_{q}^{m\times n}$ with rate $R$ and decoding radius $\rho=1-R-\epsilon$ can not be list decoded in ${\rm poly}(n)$ time. Secondly, we show that if $\frac{n}{m}$ tends to a constant $b\leq 1$, then every $F_q$-linear rank metric code in $F_{q}^{m\times n}$ with rate $R$ and list decoding radius $\rho$ satisfies the Gilbert-Varsharmov bound, i.e., $R\leq (1-\rho)(1-b\rho)$. Furthermore, for arbitrary $\epsilon>0$ and any $0<\rho<1$, with high probability a random $F_q$-linear rank metric codes with rate $R=(1-\rho)(1-b\rho)-\epsilon$ can be list decoded up to a fraction $\rho$ of rank errors with constant list size $L$ satisfying $L\leq O(\exp(1/\epsilon))$.