On the hull-variation problem of equivalent vector rank metric codes

TL;DR

This paper investigates the hull-variation problem for vector rank-metric codes, showing that all such codes over finite fields are equivalent to LCD codes, extending classical results to the rank-metric setting.

Contribution

It introduces the hull-variation problem for vector rank-metric codes and proves that all these codes over finite fields are equivalent to LCD codes, generalizing classical Hamming-metric results.

Findings

Every vector rank-metric code over any finite field is equivalent to an LCD code.

The results apply specifically when the field size q=2 or q=3.

The study extends classical hull-variation concepts to rank-metric codes.

Abstract

The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming-metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the so-called hull-variation problem formulated by Hao Chen in 2023. In this paper, we consider the analogous problem for vector rank-metric codes, along with their associated matrix codes and extended block codes. Our results include the fact that every vector rank-metric code over any finite field $\mathbb{F}_q$, in particular when $q=2$ or $q=3$, is equivalent to an LCD code.