On the hull-variation problem of equivalent vector rank metric codes
For coding theorists, this resolves the hull-variation problem for vector rank-metric codes, showing that hull dimension can always be reduced to zero, but the result is incremental as it mirrors known results for Hamming-metric codes.
The paper extends the hull-variation problem from Hamming-metric codes to vector rank-metric codes, proving that every such code over any finite field is equivalent to an LCD code.
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.