Curves of genus 3 over small finite fields

Curves of Genus 3 over Small Finite Fields

Curves of Genus 3 over Small Finite Fields

Curves of Genus 3 over Small Finite Fields

The maximal number of rational points that a (smooth, geometrically irreducible) curve of genus g over a finite field lF, of cardinality q can have, is denoted by Nq(g). The interest in this number, particularly for fixed q as a function of g, arose primarily during the last two decades from applications to error correcting codes [Lath], [T-G-Z], [vL-vdG]. A lot of results on N,(g) for fixed q ...

Nondegenerate Curves of Low Genus over Small Finite Fields

In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not nondegenerate, one over F2 and one over F3. Both ...

Twists of genus three curves over finite fields

Article history: Received 13 July 2009 Revised 10 March 2010 Available online xxxx Communicated by H. Stichtenoth MSC: 11G20 14G15 14H25 14H45 14H50 14G15 14K15