Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes

2026-05-19

AI Summary

This study investigates the variability of the Hermitian hull dimension of vector rank-metric codes under equivalence transformations and resolves the long-standing existence problem for maximum rank distance (MRD) codes with arbitrary admissible Hermitian hull dimensions. By introducing scaled trace self-dual bases, the authors construct Hermitian self-orthogonal generalized Gabidulin codes over any finite field of prime power order. Furthermore, they establish—for the first time—that, apart from a single exceptional parameter set, every vector rank-metric code is equivalent to a Hermitian linear complementary dual (LCD) code. The work fully characterizes the range of permissible Hermitian hull dimensions and provides explicit constructions of the corresponding MRD codes, thereby settling the existence question for this class of codes.

Abstract

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.

Problem

• Hermitian hull
• rank-metric codes
• MRD codes
• self-orthogonal
• hull dimension

Innovation

• Hermitian hull-variation
• scaled trace-self-dual basis
• generalized Gabidulin codes
• maximum rank distance (MRD) codes
• Hermitian self-orthogonal