The Poincaré Polynomial of a Linear Code

Abstract

We introduce the Poincaré polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. We prove that the Poincaré polynomial is a complete invariant of the code in the binary and ternary case and it is not when q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for binary Reed-Muller codes.