On List-decodability of Random Rank Metric Codes (1401.2693v2)

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 $L\leq O(\exp(1/\epsilon))$.