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To each linear code over a finite field we associate the matroid of itsparity check matrix. We show to what extent one can determine the generalizedHamming weights of the code (or defined for a matroid in general) from varioussets of Betti numbers of Stanley-Reisner rings of simplicial complexesassociated to the matroid.

Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids