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 purpose of this paper is to introduce the notion of sum-matroid and to link it to the theory of sum-rank metric codes. Sum-matroids generalize both the notions of matroid and q-matroid. We show how generalized weights can be defined for them and establish a duality for these weights analogous to Wei's one for generalized Hamming weights of linear codes, and more generally, for matroids proved by Britz et al. We associate a sum-matroid to a sum-rank metric code and the corresponding results of Martínez-Peñas for sum-rank metric codes are derived as a consequence.
<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>
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-matroids, 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.
We consider a q-analogue of abstract simplicial complexes, called q-complexes, and discuss the notion of shellability for such complexes. It is shown that q-complexes formed by independent subspaces of a q-matroid are shellable. We also outline some partial results concerning the determination of homology of shellable q-complexes.
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