Equivalence, group of automorphism and invariants of a family of rank metric codes arising from linearized polynomials

Abstract

Let Fq denote the finite field with q=pλ elements. Maximum Rank metric codes (MRD for short) are subsets of Mm×n(Fq) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) presented a new class of MRD codes over Fq called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes Hk,s(L1,L2). The equivalence and duality of twisted Gabidulin codes were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in M2n×2n(Fq) was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case L1(x)=x, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of Hk,s(x,L(x)) and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of some twisted Gabidulin codes.