Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in $\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the rank distance is called a symmetric rank-metric code. In this paper we study the covering properties of symmetric rank-metric codes. First we characterize symmetric rank-metric codes which are perfect, i.e. that satisfy the equality in the sphere-packing like bound. We show that, despite the rank-metric case, there are non trivial perfect codes. Also, we characterize families of codes which are quasi-perfect.

翻译：令 $\mathrm{Sym}_q(m)$ 表示 $\mathbb{F}_q^{m\times m}$ 中的对称矩阵空间。配备秩距离的 $\mathrm{Sym}_q(m)$ 子空间称为对称秩度量码。本文研究对称秩度量码的覆盖性质。首先，我们刻画了满足类球填充界等式的完美对称秩度量码。研究表明，与一般秩度量情形不同，存在非平凡的完美对称秩度量码。此外，本文还刻画了拟完美码的若干族类。