On symmetric and Hermitian rank distance codes
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Let M denote the set S_n, q of n × n symmetric matrices with entries in GF(q) or the set H_n, q^2 of n × n Hermitian matrices whose elements are in GF(q^2). Then M equipped with the rank distance d_r is a metric space. We investigate d-codes in ( M, d_r) 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 (3q^n-q^n/2)/2, and of a 2-code of size q^6+ q(q-1)(q^4+q^2+1)/2, for n = 3. In the symmetric case, if n is odd or if n and q are both even, 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 q^4+q^3+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.
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