Explicit bases for Riemann-Roch spaces of the extended norm-trace function field, with applications AG codes and Weierstrass semigroups

The extended norm-trace function field is a generalization of the Hermitian and norm-trace function fields which are of importance in coding theory. In this paper, we provide explicit bases for certain Riemann-Roch spaces on the extended norm-trace function field. These bases provide explicit generator and parity check matrices for algebraic geometry codes CL ( D, aP∞ + ∑ β∈B aβP0β ) on the extended norm-trace function field. This includes one-point codes as well as multipoint codes supported by any of the places P∞ or P0β. In addition, Weierstrass semigroups of m-tuples of these places on the norm-trace function field are determined via dimensions of associated Riemann-Roch spaces. Results on codes and semigroups associated with the Hermitian and norm-trace function fields are special cases of those presented here. ∗Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975 email: [email protected] †G. L. Matthews’ work was supported in part by NSF DMS-0901693 and NSA H-9823006-1-0008. ‡Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975 email: [email protected]