On polynomial invariants of codes, matroids, and the partition function

A linear code can be thought of as a vector matroid represented by the columns of code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to: (1) relate the weight polynomial of codes and the reliability polynomial of linear matroids and to prove new bounds on the latter; (2) prove that the partition polynomial of the Potts model equals the weight polynomial of the cocycle code of the underlying graph; (3) give a simple proof of Greene's theorem and its generalization