On the De Boer–Pellikaan method for computing minimum distance

If C is an [n,k,d]-linear code, computing its minimum distance, d, leads to deciding if certain ideals I generated by products of linear forms are Artinian or not (De Boer and Pellikaan, 1999, ). In this note we show that when these ideals are Artinian, then they must be powers of the maximal (irrelevant) ideal. We discuss some theoretical consequences of this result in connection to projective minimal codewords. In the end we compare the De Boer–Pellikaan method with the Migliore–Peterson method (Migliore and Peterson, 2004).