On symmetric and Hermitian rank distance codes

Let $\cal M$ denote the set ${\cal S}{n, q}$ of $n \times n$ symmetric matrices with entries in ${\rm GF}(q)$ or the set ${\cal H}{n, q^2}$ of $n \times n$ Hermitian matrices whose elements are in ${\rm GF}(q^2)$. Then $\cal M$ equipped with the rank distance $dr$ is a metric space. We investigate $d$-codes in $({\cal M}, dr)$ 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 $\cal M$, $n$ even and $n/2$ odd, of size $\left(3q^{{n}-q{n/2}\right)/2$,} and of a $2$-code of size $q⁶⁺ q(q-1)(q^4+q2+1)/2$, 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 $q^4+q3+1$ 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.