Hermitian Rank Metric Codes and Duality

Abstract

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 $\mathbb {F}_{q^{2m}}^{n}$ .