A new family of maximum linear symmetric rank-distance codes

TL;DR

This paper introduces a new family of maximum linear symmetric rank-distance codes over finite fields, specifically for dimensions 6, 8, and 10, expanding the known constructions and classifying their equivalence classes.

Contribution

It provides the first known constructions of maximum additive symmetric rank-distance codes for certain dimensions and fully characterizes their equivalence relations.

Findings

Constructed new maximum linear symmetric rank-distance codes for n=6,8,10.

Proved these codes are not equivalent to existing ones.

Classified the equivalence classes within the new family.

Abstract

Let $\mathscr{S}_n(q)$ denote the set of symmetric bilinear forms over an $n$-dimensional $\mathbb{F}_q$-vector space. A subset $\mathcal{C}$ of $\mathscr{S}_n(q)$ is called a $d$-code if the rank of $A-B$ is larger than or equal to $d$ for any distinct $A$ and $B$ in $\mathcal{C}$. If $\mathcal{C}$ is further closed under matrix addition, then $|\mathcal{C}|$ is sharply upper bounded by $q^{n(n-d+2)/2}$ if $n-d$ is even and $q^{(n+1)(n-d+1)/2}$ if $n-d$ is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum $\mathbb{F}_q$-linear $(n-2)$-codes in $\mathscr{S}_n(q)$ for $n=6,8$ and $10$ which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family.