Divisible Linear Rank Metric Codes
Olga Polverino, Paolo Santonastaso, John Sheekey, Ferdinando Zullo
Abstract
A subspace of matrices in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{q^{e}}^{m\times n}$ </tex-math></inline-formula> can be naturally embedded as a subspace of matrices in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{q}^{em\times en}$ </tex-math></inline-formula> with the property that the rank of any of its matrix is a multiple of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$e$ </tex-math></inline-formula> . It is quite natural to ask whether or not all subspaces of matrices with such a property arise from a subspace of matrices over a larger field. In this paper we explore this question, which corresponds to studying divisible codes in the rank metric. We determine some cases for which this question holds true, and describe counterexamples by constructing subspaces with this property which do not arise from a subspace of matrices over a larger field.