Quadratic forms, generalized Hamming weights of codes and curves with many points
The paper under review continues the series of the authors' papers [Compos. Math. 84, 333--367 (1992; Zbl 0804.14014), Combin. Theory Ser. A 70, 337--348 (1995; Zbl 0824.94025)]. The main idea is in an interplay between binary trace codes, quadratic forms over \(\mathbb F_2\), and fibre products of Artin-Schreier curves. The authors construct subcodes in trace codes consisting of minimum weight words. This allows them to obtain curves with many rational points. Namely, for certain genera they improve known lower bounds for the maximal number of rational points and exhibit some curves attaining the Hasse-Weil upper bound.
scientific article; zbMATH DE number 913829