Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers

TL;DR

This paper links the generalized weight spectra of Gabidulin rank-metric codes to Betti numbers of associated $q$-matroids, providing a new algebraic-combinatorial framework for analyzing these codes.

Contribution

It introduces a novel connection between $q$-matroid Betti numbers and the weight spectra of Gabidulin codes, advancing the algebraic understanding of rank-metric codes.

Findings

Generalized rank weights expressed via Betti numbers of dual classical matroids.

Weight distribution derived from $q$-matroid M"{o}bius functions.

Determination of higher weight spectra from $q$-matroid structures.

Abstract

We consider $q$-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized rank weights of a Gabidulin rank-metric code in terms of Betti numbers of the dual classical matroid associated to the $q$-matroid corresponding to the code. In our main result, we show how these Betti numbers and their elongations determine the generalized weight polynomials for $q$-matorids, in particular, for the Gabidulin rank-metric codes. In addition, we demonstrate how the weight distribution and higher weight spectra of such codes can be determined directly from the associated $q$-matroids by using M\"{o}bius functions of its lattice of $q$-flats.