On maximum additive Hermitian rank-metric codes

Abstract

Inspired by the work of Zhou (Des Codes Cryptogr 88:841–850, 2020) based on the paper of Schmidt (J Algebraic Combin 42(2):635–670, 2015), we investigate the equivalence issue of maximum d-codes of Hermitian matrices. More precisely, in the space \({{H}}_n(q^2)\) of Hermitian matrices over \({\mathbb {F}}_{q^2}\) we have two possible equivalences: 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 \({H}_n(q^2)\). 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. (Discrete Math 343(7):111871, 2020).