From Linear Codes to Hyperplane Arrangements via Thomas Decomposition

TL;DR

This paper links linear codes to hyperplane arrangements through Thomas decomposition, generalizing weight enumerators to infinite fields and refining existing code enumeration methods.

Contribution

It introduces a novel approach connecting linear codes with hyperplane arrangements using Thomas decomposition, enabling analysis over infinite fields.

Findings

Generalization of weight enumerator to infinite fields

Refinement of code enumeration via hyperplane arrangements

Unified treatment of codes over various finite fields

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.