| Summary: | In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-linear matrix codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> and for both <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula>-linear codes in the ambient space <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-duality between bases of <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula> and prove that a <inline-formula> <tex-math notation="LaTeX">$q^{m}$ </tex-math></inline-formula>-self dual basis exists if and only if <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is an odd integer. We obtain connections on the dual codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula> with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$(\mathbb {F}_{q^{2}})_{n,m}$ </tex-math></inline-formula>. Furthermore, we present connections between Hermitian <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes and Euclidean <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula>-additive codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2m}}^{n}$ </tex-math></inline-formula>.
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