2021/06/21 by Johnsen, Trygve, Pratihar, Rakhi, Verdure, Hugues · 1 citation
#05B35 #11T71 #94B60 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2106.10993
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 Möbius functions of its lattice of q-flats.