We study the Hermitian hull-variation problem for vector rank-metric codes. Except for one parameter pair, we show that the Hermitian hull dimension of such a code can be reduced to any smaller value within its equivalence class, and in particular every such code is equivalent to a Hermitian LCD code. We then address the existence of maximum rank distance (MRD) codes with prescribed Hermitian hull dimension. To this end, we introduce the notion of a \emph{scaled trace-self-dual basis} of a finite field extension, which exists in all cases, and use it to construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power. Combined with the hull-variation theorem, this yields MRD codes attaining every admissible Hermitian hull dimension.

翻译：我们研究向量秩度量码的埃尔米特壳变化问题。除了一组参数对之外，我们证明此类码的埃尔米特壳维数可以在其等价类内减小到任意更小的值，特别地，每个此类码等价于一个埃尔米特LCD码。随后，我们探讨具有指定埃尔米特壳维数的最大秩距离（MRD）码的存在性问题。为此，我们引入有限域扩张的\textit{缩放迹自对偶基}的概念，该基在所有情形下均存在，并利用它构造所有素数幂下的埃尔米特自正交广义Gabidulin码。结合壳变化定理，这给出了达到所有允许的埃尔米特壳维数的MRD码。