Title: Some Interactions between Coding Theory, Combinatorics, and Commutative Algebra
Abstract: We will outline some interactions between linear error correcting codes, combinatorial structures known as matroids, and certain notions and results in commutative and homological algebra. In particular, we will discuss a relatively recent work of Johnsen and Verdure where they associate afine set of invariants, called Betti numbers, to linear codes. These are obtained by considering certain Stanley-Reisner rings corresponding to linear codes and studying the graded minimal free resolutions of these rings. It turns out that these Betti numbers determine several important parameters of linear codes such as generalized Hamming weights and generalized weight enumerators. However, computing Betti numbers is usually a hard problem. But it is tractable if the free resolution is "pure". We will then outline an intrinsic characterization of purity of graded minimal free resolutions associated with linear codes. Further, we will discuss a characterization of (generalized) Reed-Muller and also projective Reed-Muller codes that admit a pure resolution. If time permits, we will also discuss the case of rank metric codes, which have been of some current interest, and relevant q-analogues of the notions of matroids and simplicial complexes, and the notions of shellability and homology for these objects.