Abstract. We study Algebraic Geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.

We investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincaré series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.

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We give a method to construct Locally Recoverable Error-Correcting codes. This method is based on the use of rational maps between affine spaces. The recovery of erasures is carried out by Lagrangian interpolation in general and simply by one addition in some good cases.2010 Mathematics Subject Classification. 94B27, 11G20, 11T71, 14G50, 94B05.

In this work we study the generalized Weierstrass semigroup H(P m ) at an m-tuple P m = (P 1 , . . . , P m ) of rational points on certain curves admitting a plane model of the form fIn particular, we compute the generating set Γ(P m ) of H(P m ) and, as a consequence, we explicit a basis for Riemann-Roch spaces of divisors with support in {P 1 , . . . , P m } on these curves, generalizing results of Maharaj, Matthews, and Pirsic in [14].