Weight distributions of subfield subcodes of algebraic-geometrical codes (Q809986)

In the paper some properties of subcodes of algebraic-geometric codes over prime subfields are investigated. In particular, the weighted distributions of these codes are discussed. Two theorems that describe differences between these distributions and those obtained for random codes, i.e., with binomial weight distributions, are given. It has been shown that for sufficiently long codes the weight distributions of algebraic-geometric codes become very close to appropriate weight distributions of random codes. To prove the theorems the authors use Stirling formulae and Weil-Bombieri estimations for exponential sums in finite fields. The paper brings some new, interesting results that extend earlier ones (e.g., Goppa's results) obtained in the area of algebraic- geometric codes.