Generalized Reed–Muller Codes and Curves with Many Points Original Research Article

Words of low weight in trace codes correspond to curves with many points and the same holds for subcodes of low weight via the fibre product construction. In 1996 Heijnen and Pellikaan gave an algorithm to determine a basis of minimum weight subcodes of generalized Reed–Muller codes. We show how this algorithm can be used to produce curves with many points and also some new families of curves which reach the Hasse–Weil upper bound.