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 Grobner bases. First, we give a historical overview of decoding cyclic codes via solving systems polynominal 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 Grobner 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