Conference Agenda

The problem of finding good codes is central to the theory of error correcting codes. For many years coding theorists have addressed this problem by adding algebraic and combinatorial structure to C.

In the early 80s Goppa used algebraic curves to construct linear error correcting codes, the socalled algebraic geometric codes (AG codes). The construction of an AG code with alphabet a finite field Fq requires that the underlying curve is Fq-rational and involves two Fq-rational divisors D and G on the curve.

In this minisymposium we will present results on Algebraic Geometry codes and their performances.

The Giulietti-Korchmáros (GK) curve C is a maximal curve over GF(q^6) that was discovered in 2009. The first topic that is addressed in this talk, concerns the structure of the Weierstrass semigroups of points of this curve. It turns out that there are three possibilities for these semigroups and that the Weierstrass points of the GK curve are exactly the GF(q^6)-rational points. A description of these three possible Weierstrass semigroups will be presented. The GK curve was generalized by Garcia, Stichtenoth and Xing in 2010 in the construction of the so called GGS maximal curves. More precisely they found for each odd n>2 a curve Cn, maximal over GF(q2n). The curve C3 equals the GK curve C. In the second part of this talk a different generalization of the GK curve will be presented. Similarities and differences with the GGS curves will be discussed, especially their genera and automorphism groups. This is a joint work with Peter Beelen.

For any prime power q and odd integer n≥5, we consider the F_q^2n-maximal curve X_q,n : Z^{(q^n+1)/(q+1)}=Y^{q^2}-Y, Y^q+1=X^q+X introduced by Garcia, Güneri, and Stichtenoth, and we construct over F_q^2n dual one-point AG codes C from an F_q^2-rational affine point P of X_q,n.

We study the automorphism group of C starting from the automorphism group of the curve. We determine the Weierstrass semigroup at any affine F_q^2-rational point P of X_q,n and apply this result to the parameters of C; in particular, we compute the Feng-Rao minimum distance of C when q=2. Finally, we apply some constructions known in the literature to our codes, in order to produce families of quantum codes and convolutional codes.

Algebraic Geometry codes are studied for applications as error-correcting codes, to code-based post-quantum cryptosystems and ramp secret-sharing schemes, and so on. Thus they are mathematical objects not only described on papers but to be computed explicitly on computers. As originally defined by Goppa, AG codes are based on algebraic curves and their function fields. Therefore a computing environment for AG codes should allow computations with them as well. Such a computing environment is de facto unique, and it is Magma. Though powerful, the closed nature of the software is an obstacle in spreading the achievements of researchers in this field to other researchers and students. Here we demonstrate the current status of the endeavors to provide an open source computing environment on Sage for algebraic curves, function fields, and AG codes.

This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters [216,190,>= 18] over GF(64) yields a new record.