Homological methods applied to theory of codes and matroids

Abstract

In this thesis we first give a survey of linear error-correcting codes, and how many of their most important properties only depend on the matroids derived from their parity check matrices. We also introduce the Stanley-Reisner ring associated to the simplicial complex of the independent sets of a matroid.

We then recall in particular how some important properties of linear codes, including their generalized weight polynomials, are dependent only on the Z-graded Betti numbers for the Stanley-Reisner rings of their associated matroids, and the so-called elongations of these matroids. We will use this fact to find the generalized weight polynomials of simplex codes and Reed-Muller codes of the first order.