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

TL;DR

This paper investigates the Hermitian hull-variation of vector rank-metric codes, showing most can be reduced to Hermitian LCD codes and constructing MRD codes with prescribed Hermitian hull dimensions.

Contribution

It introduces the scaled trace-self-dual basis for finite fields and constructs Hermitian self-orthogonal Gabidulin codes for all prime powers, enabling MRD codes with any admissible Hermitian hull dimension.

Findings

Hermitian hull dimension can be reduced within equivalence classes.

Every code is equivalent to a Hermitian LCD code except for one parameter pair.

Constructed MRD codes with all admissible Hermitian hull dimensions.

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.