On maximum additive Hermitian rank-metric codes
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On maximum additive Hermitian rank-metric codes Rocco Trombetti1 · Ferdinando Zullo2 Received: 18 January 2020 / Accepted: 13 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Hn (q 2 ) of Hermitian matrices over Fq 2 we have two possible equivalences: the classical one coming from the maps that preserve the rank in Fqn×n 2 , and the one that comes from restricting to those maps preserving both the rank and the space Hn (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). Keywords Hermitian matrix · Rank metric code · Linearized polynomial Mathematics Subject Classification 05E15 · 05E30 · 51E22
This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). The last author was also supported by the Project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.
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Ferdinando Zullo [email protected] Rocco Trombetti [email protected]
1
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, 80126 Naples, Italy
2
Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, Viale Lincoln, 5, 81100 Caserta, Italy
123
Journal of Algebraic Combinatorics
1 Introduction Let us consider Fqn×n , the set of the square matrices of order n defined over Fq , with q a prime power. It is well known that Fqn×n equipped with d(A, B) = rk(A − B), where A, B ∈ Fqn×n , is a metric space. If C is a subset of Fqn×n with the property that for each A, B ∈ C then d(A, B) ≥ d with 1 ≤ d ≤ n, then we say that C is a d-code. Furthermore, we say that C is additive if C is an additive subgroup of (Fqn×n , +), and C is Fq -linear if C is an Fq -subspace of (Fqn×n , +, ·), where + is the classical matrix addition and · is the scalar multiplication by an element of Fq . Delsarte [10] shows the following bound for a d-code C |C| ≤ q n(n−d+1) , known as Singleton like bound, see also [12]. Codes whose parameters satisfy the aforementioned bound are known as maximum rank distance codes (or shortly MRDcodes), and they have several important applications. Attention has been paid also to rank metric codes with restrictions, which a
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