Intersections of Algebraic Curves and their link to the weight enumerators of Algebraic-Geometric Codes

Cycle 31th Oral Defence of the Phd Thesis

February 11, 2019
Versione stampabile

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 Hasse-Weil 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 algebraic-geometric 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 one-point AG-codes arising from the GK-curve. 
Afterwards we will focus our attention on the study the possible intersections between the Norm-Trace 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 one-point code. 

Supervisors: Massimiliano Sala - Giancarlo Rinaldo