Conference Agenda

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Session Overview
Session
MS185, part 3: Algebraic Geometry Codes
Time:
Saturday, 13/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-122
52 seats, 100m^2

Presentations
10:00am - 12:00pm

Algebraic Geometry Codes

Chair(s): Daniele Bartoli (Univerity of Perugia, Italy), Anna-Lena Horlemann (University of St. Gallen)

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.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Subcovers and codes on a class of trace-defining curves

Guilherme Tizziotti
Federal University of Uberlandia

In this work, we construct some class of explicit subcovers of the curve Xn,r defined over Fq^n by affine equation yq^(n-1)+...+yq+y=xq^(n-r)+1-xq^n+q^(n-r). These subcovers are defined over Fq^n by affine equation gs(y)=xq^n+q^(n-r)-xq^(n-r)+1, where gs(y) is a q-polynomial of degree qs. The Weierstrass semigroup H(P), where P is the only point at infinity on such subcovers, is determined for 1 ≤ s ≤ 2r-n+1 and the corresponding one-point AG codes are investigated. Codes establishing new records on the parameters with respect to the previously known ones are discovered, and 108 improvements on MinT tables are obtained.

 

On Weierstrass semigroup at $m$ points on curves of the form $f(y)=g(x)$

Alonso SepĂșlveda Castellanos
Federal University of Uberlandia

In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain m points on curves X with plane model of the form f(y)=g(x) over Fq, where f(T),g(T) in Fq[T]. Our results were obtained using the concept of discrepancy, for given points P and Q on X. This concept was introduced by Duursma and Park, and allows us to make a different and more general approach than that used to certain specific curves studied earlier.

 

Pure gaps on curves with many rational places

Ariane Masuda
NYC College of Technology

We consider the algebraic curve defined by ym=f(x) where m≥2 and f(x) is a rational function over Fq. 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.

 

Non projective Frobenius algebras and linear codes

Javier Lobillo Borrero
Universidad de Granada

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.