Let ${\mathbb{F}}_q$ denote the finite field with $q=p^{\lambda }$ elements. Maximum Rank metric codes (MRD for short) are subsets of ${\mathrm{M}}_{m\times n}({\mathbb{F}}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) 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 were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in ${\mathrm{M}}_{2n\times 2n}({\mathbb{F}}_q)$ 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 some twisted Gabidulin codes.

让 ${\mathbb{F}}_q$ 用 $q={磷}^{\lambda }$元素。最大秩度量代码（简称 MRD）是${\mathrm{米}}_{米\times n}({\mathbb{F}}_q)$其元素数量达到类似单例的界限。Delsarte (1978) 和 Gabidulin (1985) 发现了第一个已知的 MRD 代码。Sheekey (2015) 提出了一类新的 MRD 代码${\mathbb{F}}_q$ 称为扭曲 Gabidulin 代码，并提出了将扭曲 Gabidulin 代码推广到这些代码 ${\mathcal{H}}_{克,秒}({升}_1,{升}_2)$. Lunardon、Trombetti 和 Zhou (2016) 讨论了扭曲 Gabidulin 代码的等价性和对偶性。一类新的 MRD 代码${\mathrm{米}}_{2n\times 2n}({\mathbb{F}}_q)$由 Trombetti-Zhou (2018) 发现。在这项工作中，我们描述了 Sheekey 提出的代码类的等价性，概括了已知的扭曲 Gabidulin 代码和 Trombetti-Zhou 代码的结果。在论文的第二部分，我们将自己限制在案例中${升}_1(X)=X$，我们展示了它的右核、中核、Delsarte 对偶和伴随代码。在最后一节中，我们提出了自同构群${\mathcal{H}}_{克,秒}(X,升(X))$并计算其基数。特别地，我们获得了一些扭曲的 Gabidulin 代码的自同构群中的元素数。