Topics in Finite Fields: cyclic codes, Artin-Schreier's hypersufarces and irredu...
Topics in Algebraic Curves: Zeta Function and Frobenius nonclassical curves
Geometry of manifolds in the euclidian space and in the Minkowski space
| Full text | |
| Author(s): |
Martinez, F. E. Brochero
;
Oliveira, Daniela
Total Authors: 2
|
| Document type: | Journal article |
| Source: | DESIGNS CODES AND CRYPTOGRAPHY; v. 92, n. 10, p. 22-pg., 2024-06-05. |
| Abstract | |
Let Fqn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q<^>n}$$\end{document} represent the finite field with qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>n$$\end{document} elements. In this paper, our focus is on determining the number of Fqn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q<^>n}$$\end{document}-rational points for two specific objects: an affine Artin-Schreier curve given by the equation yq-y=x(xqi-x)-lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<^>q-y = x(x<^>{q<^>i}-x)-\lambda $$\end{document}, and an Artin-Schreier hypersurface given by the equation yq-y=& sum;j=1rajxj(xjqij-xj)-lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y<^>q-y=\sum _{j=1}<^>r a_jx_j(x_j<^>{q<^>{i_j}}-x_j)-\lambda $$\end{document}. Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element lambda is an element of Fqn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {F}_{q<^>n}$$\end{document} over the subfield Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} is equal to zero. (AU) | |
| FAPESP's process: | 22/14004-7 - Topics in Finite Fields: cyclic codes, Artin-Schreier's hypersufarces and irreducible trinomials |
| Grantee: | Daniela Alves de Oliveira |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |