Topics in Finite Fields: cyclic codes, Artin-Schreier's hypersufarces and irreducible trinomials

• Grant number: 22/14004-7
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2023
• Status: Discontinued
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Herivelto Martins Borges Filho
• Grantee: Daniela Alves de Oliveira
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 25/02198-0 - Topics in Finite Fields and Combinatorics, BE.EP.PD

Abstract

This project has two main objectives: the first is to determine the number of rational points of Artin-Schreier hypersurfaces, which are hypersurfaces of the form y^q-y=f(X), with f(X) \in F_q[X]{0} and X=(x_1,...,x_r), imposing some conditions on f. This problem has already been partially studied in my Ph.D thesis in the case where f is a sum of binomials of the form a_j x_j(x_j^{q^{i_j}}+x_j).We will associate this type of hypersurface with multidimensional cyclic codes and study these codes to determine their minimum weight and their weight enumerator. The second objective is to study trinomials and pentanomials over F_2, in the direction of getting irreducibility and existence conditions, and still general results that characterize families of irreducible trinomials/pentanomials. The existence and construction of irreducible polynomials with few monomials over F_2[x] is of great theoretical interest and has applications in cryptography and code theory, among others.The problems proposed in this project are continuations or generalizations of problems studied in my Ph.D. These problems are of interest to the scientific community, not only for the theoretical aspect, but also for their several applications in another areas. During the post-doctorate, we will deepen our study of tools necessary to obtain results in these problems, such as: Algebraic Geometry, Character Theory, Quadratic Forms, among others. In addition, we will review the results associated with these problems, exploring the techniques and notions already developed for the approach of general problems inserted in this context.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MARTINEZ, F. E. BROCHERO; OLIVEIRA, DANIELA. On the number of rational points of Artin-Schreier's curves and hypersurfaces. DESIGNS CODES AND CRYPTOGRAPHY, v. 92, n. 10, p. 22-pg., 2024-06-05. (22/14004-7)