On List-Decodability of Random Rank Metric Codes and Subspace Codes

Abstract

Codes in rank metric have a wide range of applications. To construct such codes with better list-decoding performance explicitly, it is of interest to investigate the listdecodability of random rank metric codes. It is shown that if n/m = b is a constant, then for every rank metric code in Fm×n q with rate R and list-decoding radius ρ must obey the Gilbert-Varshamov bound, that is, R ≤ (1-ρ)(1-bρ). Otherwise, the list size can be exponential and hence no polynomial-time list decoding is possible. On the other hand, for arbitrary 0 <; ρ <; 1 and E > 0, with E and ρ being independent of each other, with high probability, a random rank metric code with rate R = (1 - ρ)(1 - bρ) - can be efficiently list-decoded up to a fraction ρ of rank errors with constant list size O(1/E). We establish similar results for constant-dimension subspace codes. Moreover, we show that, with high probability, the list-decoding radius of random Fq-linear rank metric codes also achieve the Gilbert-Varshamov bound with constant list size O(exp(1/E)).