On perfect symmetric rank-metric codes

Let \(\textrm{Sym}_q(m)\) be the space of symmetric matrices in \({\mathbb {F}}_q^{m\times m}\). A subspace of \(\textrm{Sym}_q(m)\) equipped with the rank distance is called an \({{\mathbb {F}}}_{q}\)-linear symmetric rank-metric code. In this paper, we study the covering properties of \({{\mathbb {F}}}_{q}\)-linear symmetric rank-metric codes. First we characterize \({{\mathbb {F}}}_{q}\)-linear 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. Indeed, we prove that the only perfect non-trivial \({{\mathbb {F}}}_{q}\)-linear symmetric rank-metric codes in \(\textrm{Sym}_q(m)\) are the symmetric MRD codes with minimum distance 3 and m odd. Also, we characterize families of codes which are quasi-perfect.