Hermitian rank distance codes

Let $$X=X(n,q)$$X=X(n,q) be the set of $$n\times n$$n×n Hermitian matrices over $$\mathbb {F}_{q^2}$$Fq2. 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 $$A,B\in Y$$A,B∈Y, the rank of $$A-B$$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 $$4\le d\le n-2$$4≤d≤n-2, constructions of d-codes are given, which are optimal among the d-codes that are subgroups of $$(X,+)$$(X,+). This work complements results previously obtained for several other types of matrices over finite fields.

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