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.

We consider the algebraic curve defined by y^m=f(x) where m≥2 and f(x) is a rational function over F_q. We extend the concept of pure gap to c-gap and obtain a criterion to decide when an s-tuple is a c-gap at s rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places. We present the parameters of codes constructed using our families of pure gaps. This is joint work with Bartoli, Montanucci, and Quoos.

We extend the notion of a Frobenius algebra, dropping the projectivity condition, to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one’s options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the McWilliam identities that may be obtained in this context.