Venue: Seminar Room “1” – Department of Mathematics – Via Sommarive, 14 Povo  Trento
Time: 10.30 a.m.

Matteo Bonini  PhD in Mathematics
Abstract:
Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the HasseWeil upper bound for the number of rational points compared with the genus of the curve.
In most cases, the weight distribution of a given code is hard to be computed. Even the problem of computing codewords of minimum weight can be a difficult task, apart from specific cases.
In this seminar we investigate the general problem of determining the minimum distance and the weight enumerator polynomial of algebraicgeometric codes. We adopt an approach based on the intersection of the curve defining the code and other curves.
We compute the maximal number of intersections that the GK curve can have with a plane curve of degree lower or equal to three. These results are used to determine the minimum distance and the number of minimum weight codewords of dual onepoint AGcodes arising from the GKcurve.
Afterwards we will focus our attention on the study the possible intersections between the NormTrace curve and a parabola. We obtain sharp bounds for the spectrum of the intersections between these two curves. For this purpose, we translate the problem of finding the intersections between these two curves into that of determining the rational points of a certain cubic surfaces. To do so, we employ techniques coming from the properties of irreducible cubic surfaces over finite fields. Then we partially deduce information on the weight enumerator polynomial of the corresponding onepoint code.
Supervisors: Massimiliano Sala  Giancarlo Rinaldo