On symmetric and Hermitian rank distance codes

Let Mdenote the set S-n,S-q of n x n symmetric matrices with entries in F-q or the set H-n,H-q2 of n x n Hermitian matrices whose elements are in F-q2. Then Mequipped with the rank distancedris 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, neven 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 nis 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 q(4) + q(3) + 1is 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]. (C) 2022 Elsevier Inc. All rights reserved.