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arXiv · 2406.12450

On perfect symmetric rank-metric codes

Usman Mushrraf·Ferdinando Zullo
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Abstract

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.

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Usman Mushrraf, Ferdinando Zullo. 2024-06-18. On perfect symmetric rank-metric codes. https://arxiv.org/abs/2406.12450

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