Explicit Construction of Good Towers of Function Fields Nhut Nguyen - PowerPoint PPT Presentation

Explicit Construction of Good Towers of Function Fields Nhut Nguyen Advisor: Prof. Peter Beelen joint work with Alp Bassa Technical University of Denmark

Motivation

Asymptotic Code Bounds ( N → ∞ ) Figure: For squares q ≥ 49 , there exists Algebraic Geometric Codes over GF ( q ) beating Gilbert-Varshamov Bound in some interval.

Goal Explicit construction of algebraic curves having many points over a finite field. Result A procedure (algorithm) to produce explicit equations for such good sequences of curves. Method Drinfeld modular theory . x 4 n +1 x 3 n + x 4 n +1 x 2 n + x 4 n +1 x n + x 3 n +1 x 2 n + x 3 n +1 x n + x 3 n +1 (1) + x 2 n +1 x n + x 2 n +1 + x n +1 x 3 n + x n +1 + x 4 n = 0 . Figure: An example of our optimal tower over GF (16).

Obrigado! SPCodingSchool BBQ, Campinas, Brazil, 23 Jan 2015.