Optimal Curves of Genus 3 over Finite Fields with Discriminant -19

E. Alekseenko, S. Aleshnikov, N. Markin, A. Zaytsev

Published 2009-02-11, updated 2011-08-14Version 2

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a table of maximal and minimal curves over finite fields with discriminant -19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.