On the list decodability of self-orthogonal rank-metric codes

Guruswami and Resch proved that a random Fq-linear rank-metric code is list decodable with list decoding radius attaining the Gilbert-Varshamov bound [8]. Furthermore, in Hamming metric, random linear self-orthogonal codes can be list decoded up to the Gilbert-Varshamov bound with polynomial list size [11]. Motivated by these two results and the potential applications of self-orthogonal rank-metric codes in network coding and cryptography [20], [18] and [5], we focus on investigating their list decodability. In this paper, we prove that with high probability, a random Fq-linear self-orthogonal rank-metric code over Fqn×m can be list decoded up to the Gilbert-Varshamov bound with polynomial list size. In addition, we show that an Fqm-linear self-orthogonal rank-metric code of rate up to the Gilbert-Varshamov bound with exponential list size.