On symmetric and Hermitian rank distance codes

Abstract

Let M denote the set Sn,q of n×n symmetric matrices with entries in Fq or the set Hn,q2 of n×n Hermitian matrices whose elements are in Fq2. Then M equipped with the rank distancedr is a metric space. We investigate d–codes in (M,dr) and construct d–codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an n–code of M, n even and n/2 odd, of size (3qn−qn/2)/2, and of a 2–code of size q6+q(q−1)(q4+q2+1)/2, for n=3. In the symmetric case, if n is odd, we provide better upper bound on the size of a 2–code. In the case when n=3 and q>2, a 2–code of size q4+q3+1 is exhibited. This provides the first infinite family of 2–codes of symmetric matrices whose size is larger than the largest possible additive 2–code and an answer to a question posed in [25, Section 7], see also [23, p. 176].