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Generalized Reed–Muller Codes and Curves with Many Points

✍ Scribed by G van der Geer; M van der Vlugt

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
236 KB
Volume
72
Category
Article
ISSN
0022-314X

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✦ Synopsis

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.

1998 Academic Press

In the quest for curves over finite fields with many points coding theory has been a useful guide, as words of low weight in trace codes correspond to Artin Schreier curves with many points. This correspondence can be extended to subcodes of low weight and fibre products of Artin Schreier curves. Subcodes of minimum weight of a code C determine the weight hierarchy of C and knowledge of the weight hierarchy indicates where curves with many points are likely to be found. However, determination of weight hierarchies is a hard problem in coding theory. In 1990 Wei found the weight hierarchy of the classical binary Reed Muller codes (see [W]). Six years later Heijnen and Pellikaan succeeded in finding the weight hierarchy of Reed Muller codes over arbitrary finite fields, cf.

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