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We consider $q$-matroids and their associated classical matroids derived fromGabidulin rank-metric codes. We express the generalized rank weights of aGabidulin rank-metric code in terms of Betti numbers of the dual classicalmatroid associated to the $q$-matroid corresponding to the code. In our mainresult, we show how these Betti numbers and their elongations determine thegeneralized weight polynomials for $q$-matorids, in particular, for theGabidulin rank-metric codes. In addition, we demonstrate how the weightdistribution and higher weight spectra of such codes can be determined directlyfrom the associated $q$-matroids by using M\"{o}bius functions of its latticeof $q$-flats.

Weight Spectra of Gabidulin Rank-metric Codes and Betti Numbers