A note on rank-metric codes

Let $\mathbb{F}_q$ denote the finite field with $q=p^\lambda$ elements. Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes known was found by Delsarte (1978) and Gabidulin (1985). Sheekey (2016) presented a new class of MRD codes over $\mathbb{F}q$ called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes $\mathcal{H}_{k,s}(L_1,L_2)$. The equivalence and duality of twisted Gabidulin codes was discussed by Lunardoni, Trombetti, and Zhou (2018). A new class of MRD codes in $M_{2n\times 2n}(\mathbb{F}_q)$ was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case $L_1(x)=x$, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of $\mathcal{H}_{k,s}(x,L(x))$ and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of the twisted Gabidulin codes.