On equivalence of maximum additive symmetric rank-distance codes

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.