Decoding and Finding the Minimum Distance with Gröbner Bases: History and New Insights

In this chapter we discuss decoding techniques and finding the minimum distance of linear codes with the use of Gröbner bases. First we give a historical overview of decoding cyclic codes via solving systems of polynomial equations over finite fields. In particular we mention papers of Cooper, Reed, Chen, Helleseth, Truong, Augot, Mora, Sala and others. Some structural theorems that use Gröbner bases in this context are presented. After that we shift to the general situation of arbitrary linear codes. We give an overview of approaches of Fitzgerald and Lax. Then we introduce our method of decoding linear codes that reduces this problem to solving a system of quadratic equations. We discuss open problems and future research possibilities