Hamming weights and Betti numbers of Stanley-Reisner rings associated to   matroids
  To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid. 
Trygve Johnsen, Hugues Verdure
math.CO