On symmetric and Hermitian rank distance codes

Abstract

Let $\mathcal{M}$ denote the set ${\mathcal{S}}_{n,q}$ of $n\times n$ symmetric matrices with entries in $\mathrm{GF}(q)$ or the set ${\mathcal{H}}_{n,q^2}$ of $n\times n$ Hermitian matrices whose elements are in $\mathrm{GF}(q^2)$. Then $\mathcal{M}$ equipped with the rank distance $d_r$ is a metric space. We investigate $d$-codes in $(\mathcal{M},d_r)$ 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 $\mathcal{M}$, $n$ even and $n/2$ odd, of size $\left(3q^n-q^{n/2}\right)/2$, 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.