On perfect symmetric rank-metric codes

TL;DR

This work studies symmetric rank-metric codes within $\mathrm{Sym}_q(m)$ under the rank distance, aiming to characterize codes that meet the sphere-packing bound (perfect codes) and to analyze covering properties via quasi-perfect families. It derives explicit sphere/ball size bounds and uses them to establish a precise classification: the only nontrivial perfect codes are symmetric MRD codes with $m$ odd and minimum distance $d=3$, aside from the trivial full-space code. The paper also reviews constructions based on linearized polynomials that yield symmetric MRD codes, notably the family $\mathcal{S}_{q,m,d}$, which is perfect when $(m,d)=(\text{odd},3)$. Finally, it investigates covering density, showing quasi-perfect families occur only for $d=1$ or $d=3$ with specific parity restrictions on $m$, and do not exist for larger half-distance parameters, thereby clarifying the landscape of perfect and quasi-perfect symmetric rank-metric codes and guiding their use in applications requiring optimal covering properties in matrix spaces.

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.

Theorems & Definitions (28)

• Theorem 1.1: schmidt2015symmetric
• Definition 1.2
• Theorem 2.1: carlitz1954sym
• Proposition 2.2
• proof
• Remark 2.3
• Theorem 3.1: Sphere-packing bound
• proof
• Definition 3.2
• Lemma 3.3