Hermitian Rank Metric Codes and Duality

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