Hermitian Rank Metric Codes and Duality
Javier de la Cruz, Jorge Robinson Evilla, Ferruh Özbudak · IEEE Access · 2021
In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for$\mathbb {F}_{q^{2}}$-linear matrix codes in the ambient space$(\mathbb {F}_{q^{2}})_{n,m}$and for both$\mathbb {F}_{q^{2}}$-additive codes and$\mathbb {F}_{q^{2m}}$-linear codes in the ambient space$\mathbb {F}_{q^{2m}}^{n}$. Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of$q^{m}$-duality between bases of$\mathbb {F}_{q^{2m}}$over$\mathbb {F}_{q^{2}}$and prove that a$q^{m}$-self dual basis exists if and only if$m$is an odd integer. We obtain connections on the dual codes in$\mathbb {F}_{q^{2m}}^{n}$and$(\mathbb {F}_{q^{2}})_{n,m}$with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in$\mathbb {F}_{q^{2m}}^{n}$and$(\mathbb {F}_{q^{2}})_{n,m}$. Furthermore, we present connections between Hermitian$\mathbb {F}_{q^{2}}$-additive codes and Euclidean$\mathbb {F}_{q^{2}}$-additive codes in$\mathbb {F}_{q^{2m}}^{n}$.