Abstract. A little over a decade ago, Johnsen and Verdure introduced a fine set of invariants, called Betti numbers, for a linear code, and more generally, a matroid. These are obtained from graded minimal free resolutions of Stanley-Reisner rings of simplicial complexes associated to the given linear code (or matroid). This association is further facilitated by the fact that simplicial complexes corresponding to codes, and more generally, matroids, are shellable. The shellability also leads to the determination of homology of these simplicial complexes. It turns out that the Betti numbers of a linear code determine several of its important parameters 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 thus 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. We then turn to 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 their shellability and homology.  We will outline some recent results and questions in this direction. 

This talk is mainly based on joint works with (i) Prasant Singh, (ii) Trygve Johnsen, (iii) Rati Ludhani, (iv) Rakhi Pratihar and Tovohery Randrianarisoa, and (v) Rakhi Pratihar, Tovohery Randrianarisoa, Hugues Verdure and Glen Wilson.