A relatively recent work of Johnsen and Verdure associates a fine 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. This association is further facilitated by the fact that simplicial complexes corresponding to codes and more generally, matroids are shellable. 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 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 may 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 their shellability and homology.