Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes
This work provides a complete solution to the Hermitian hull-variation problem for vector rank-metric codes and constructs MRD codes with arbitrary Hermitian hull dimensions, which is important for coding theory and its applications.
The paper solves the Hermitian hull-variation problem for vector rank-metric codes, showing that the Hermitian hull dimension can be reduced to any smaller value within its equivalence class, and every such code is equivalent to a Hermitian LCD code. It also constructs Hermitian self-orthogonal generalized Gabidulin codes for every prime power, yielding MRD codes with every admissible Hermitian hull dimension.
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.