A new family of maximum linear symmetric rank-distance codes
Wei Tang, Yue Zhou
TL;DR
The paper introduces a new family of maximum additive symmetric (n−2)-codes in the symmetric matrix space S_n(q) for n=6,8,10 with odd q, built from twisted q-polynomials. It proves maximality by detailed Dickson-matrix analysis, showing any nonzero codeword has rank at least n−2, and establishes that these codes are not equivalent to previously known constructions. The authors also classify equivalence relations within the newly constructed family and discuss duals and related code properties, including relations to CSRD codes. The work suggests potential generalization to all even n≥6 and highlights the significance of such MSRD codes in symmetric rank-metric coding theory.
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.