On maximum additive Hermitian rank-metric codes

Abstract: Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the equivalence issue of maximum $d$-codes of Hermitian matrices. More precisely, in the space $\mathrm{H}n(q^2)$ of Hermitian matrices over $\mathbb{F}{q^2}$ we have two possible equivalence: the classical one coming from the maps that preserve the rank in $\mathbb{F}_{q^2}^{n\times n}$, and the one that comes from restricting to those maps preserving both the rank and the space $\mathrm{H}_n(q^2)$. We prove that when $d<n$ and the codes considered are maximum additive $d$-codes and $(n-d)$-designs, these two equivalence relations coincide. As a consequence, we get that the idealisers of such codes are not distinguishers, unlike what usually happens for rank metric codes. Finally, we deal with the combinatorial properties of known maximum Hermitian codes and, by means of this investigation, we present a new family of maximum Hermitian $2$-code, extending the construction presented by Longobardi et al. in "Automorphism groups and new constructions of maximum additive rank metric codes with restrictions" (2020).