Towers of Betti Numbers of Matroids and Weight Distribution of Linear Codes and their Duals

Abstract

The main notion behind the study of matroids is linear dependence. In this thesis, we give a survey of the concepts and properties of linear error-correcting codes over finite fields being dependent only on the matroids derived from these codes. In particular, the weight distributions of linear codes, and their extensions, over bigger fields are only dependent on the N-graded Betti numbers of these matroids and their so-called elongations. We will use this fact to find the weight distributions for some important codes as constant weight codes and Hamming codes. In addition, the connection between the Betti tower of a matroid and its dual tower will be studied for general matroids.