On the spectrum for the genera of maximal curves over small fields

TL;DR

This paper investigates the possible genera of maximal curves over small finite fields, providing a complete characterization for the case when the field size is 49, and extending previous computational results.

Contribution

The paper determines the spectrum of genera for maximal curves over finite fields of order q^2 for 7 ≤ q ≤ 16, especially fully characterizing the case q=7.

Findings

Complete determination of the spectrum for q=7^2

Extension of previous computational results

New insights into the structure of maximal curves over small fields

Abstract

Motivated by previous computations in Garcia, Stichtenoth and Xing (2000) paper ,we discuss the spectrum $\mathbf{M}(q^2)$ for the genera of maximal curves over finite fields of order $q^2$ with $7\leq q\leq 16$. In particular, by using a result in Kudo and Harashita(2016) paper, the set $\mathbf{M}(7^2)$ is completely determined.