A commutative algebra approach to linear codes

Recently some methods have been proposed to find the distance and weight distribution of cyclic codes using Gröbner bases. We identify a class of codes for which these methods can be generalized. We show that this class contains all interesting linear codes and we provide variants and improvements. This approach sometimes reveals an unexpected algebraic structure in the code. We also investigate the decoding for a subclass, proving the existence of general error locator polynomials.