Hermitian rank distance codes

Abstract

Let $$X=X(n,q)$$ be the set of $$n\times n$$ Hermitian matrices over $$\mathbb {F}_{q^2}$$ . It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct $$A,B\in Y$$ , the rank of $$A-B$$ is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and $$4\le d\le n-2$$ , constructions of d-codes are given, which are optimal among the d-codes that are subgroups of $$(X,+)$$ . This work complements results previously obtained for several other types of matrices over finite fields.