On equivalence of maximum additive symmetric rank-distance codes

Abstract

Let ${\mathcal {S}}_m({\mathbb {F}})$ denote the set of symmetric matrices over a finite field ${\mathbb {F}}$. Let ${\mathcal {C}}$ be an additive subset of ${\mathcal {S}}_m({\mathbb {F}})$ which has minimum rank-distance d and $|{\mathcal {C}}|$ meets the upper bound. We call ${\mathcal {C}}$ a maximum additive d-code. In particular, when $d=m$, ${\mathcal {C}}$ is equivalent to a symplectic semifield spreadset. For a given invertible matrix P, $S_0\in {\mathcal {S}}_m({\mathbb {F}})$ and nonzero $a\in {\mathbb {F}}$, it is easy to see that the map $M \mapsto aP^T M^\sigma P + S_0$ defines an isometry on ${\mathcal {S}}_m({\mathbb {F}})$ with respect to the rank-metric. However, by examples of symplectic semifields, it is already known that there exist other maps preserving the rank-distance of maximum additive m-codes. In this paper, we prove that this can never happen when $d<m$. We also present a new construction of maximum additive 2-codes in ${\mathcal {S}}_m({\mathbb {F}})$ for odd m which are not t-designs.