Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers

Abstract

We consider q-matroids and their associated classical matroids derived from Gabidulin rank-metric codes. We express the generalized weights of a Gabidulin rank-metric code in terms of Betti numbers associated to the dual classical matroid coming from 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 Möbius functions of its lattice of q-flats.

The Projectivization Matroid of a $q$-Matroid

“…In [13], Johnsen and co-authors showed that a q-matroid M with groundspace E induces a matroid P (M) with groundset the projective space of E. This induced matroid, called the projectivization matroid of M turns out to be an interesting object to study. In fact, it was shown in that same paper, that the projectivization preserves the flat structure of M. It therefore becomes a useful tool when studying properties of q-matroids that depend only on flats.…”

“…Theorem 3.1. ( [13,Def.14,Prop. 15]) Let M = (E, ρ) be a q-matroid and let r : 2 PE → N 0 such that for all S ⊆ PE, r(S) = ρ( S ).…”

“…It was shown later on that matrix linear rank metric codes induce a q-polymatroid, a generalization of q-matroids (see [9,10]). Since then, many results on q-(poly)matroids, and how they relate to rank metric codes, have been established, see for example [2,5,6,8,9,10,13]. Because of their q-analogue nature, many of the newly discovered properties of q-matroids turn out to be analogues of well established matroid theory results.…”

“…In [13], Johnsen and co-authors make the connection between matroids and q-matroids more apparent by showing that a q-matroid induces a matroid, called the projectivization matroid. Furthermore, they show that the lattice of flats of the q-matroid is isomorphic to the lattice of flats of its projectivization matroid.…”

“…In this paper, we further investigate the construction introduced in [13] and use the projectivization matroid as a tool to study maps between q-matroids and the characteristic polynomial of a qmatroid. We define and study properties of the projectivization matroid in Section 3.…”