On perfect symmetric rank-metric codes
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.12450
- arXiv:
- arXiv:2406.12450
- Bibcode:
- 2024arXiv240612450M
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Combinatorics