Residual representation of algebraic-geometric codes.
Reinhold Hübl · 2001
Dedicated to T. Winiarski on the occasion of his sixtieth birthday. Abstract. In this paper residues and duality theory for curves are used to construct error-correcting codes and to find estimates for their parameters. Results of Goppa are extended to singular curves. In an attempt to find long codes with good parameters, algebraic-geometric codes have been introduced by Goppa [4]. The construction is based on the (residual) evaluation of the space of sections of a line bundle on a non-singular curve over a finite field at the points in the support of a given divisor. The explicit construction and description of these codes faces two major difficulties: i) To get long codes with good parameters it is necessary to construct nonsingular curves of high genus (due to the Hasse-Weil-Serre resp. the Drinfeld-Vladut bound). ii) To make this codes explicit and computable, it is necessary to determine the global space of sections of line bundles on curves. Both tasks are in general not easy to achieve and require deep insights into