From Linear Codes to Hyperplane Arrangements via Thomas Decomposition

Abstract

We establish a connection between linear codes and hyperplane arrangements using the Thomas decomposition of polynomial systems and the resulting counting polynomial. This yields both a generalization and a refinement of the weight enumerator of a linear code. In particular, one can deal with infinitely many finite fields simultaneously by defining a weight enumerator for codes over infinite fields.

This has been rejected since it contains some mistakes. A similar work has been publish in Documenta Math. vol. 19