Rank metric codes and their Galois duality

In this paper, we describe the Galois dual of rank metric codes in ambient space $F_Q^{n\times m}$ and $F_{Q^m}^n$, where Q=qe. We obtain connections between duality with respect to distinct inner products. Furthermore, for 0 ≤ s < e, introduce concept qsm-dual bases FQm over FQ some conditions about existence qsm-self-dual basis.