On perfect symmetric rank-metric codes
<|reference_start|>On perfect symmetric rank-metric codes: 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.<|reference_end|>
@article{mushrraf2024on,
    title={On perfect symmetric rank-metric codes},
    author={Usman Mushrraf and Ferdinando Zullo},
    journal={arXiv preprint arXiv:2406.12450},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.12450},
    primaryClass={cs.IT math.CO math.IT}
}