Stanley-Reisner rings associated to matroids and codes

To large classes of error-correcting block codes, including almost affine codes (and therefore linear codes) there are associated matroids. Many of the most useful and important properties of these codes are determined by the structures of these matroids. Moreover these structures can be read off from the homological properties of the Stanley-Reisner rings of these matroids, viewed as simplicial complexes via their independent sets. We will sketch how the code parameters, in particular the generalized Hamming weights of the codes, and also the weight polynomials, which determine the number of codewords of specified weight of extensions of the original alphabet, are determined by certain Betti number of the Stanley Reisner rings in question.