On symmetric and Hermitian rank distance codes

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