Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers [article]

Trygve Johnsen, Rakhi Pratihar, Hugues Verdure
2021 arXiv   pre-print
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
more » ... te 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.
arXiv:2106.10993v3 fatcat:rvl6djeniranjeldmrlo7wriry