Uniqueness of low genus optimal curves over F_2

TL;DR

This paper investigates the uniqueness of optimal algebraic curves of low genus over the finite field F_2, establishing uniqueness for genus up to 5 and identifying multiple optimal curves for genus 6 and 7.

Contribution

It proves the uniqueness of optimal curves over F_2 for genus g ≤ 5 and identifies the number of such curves for genus 6 and 7, advancing understanding of their classification.

Findings

Unique optimal curves for g ≤ 5 over F_2

Two optimal curves for g=6 over F_2

At least two optimal curves for g=7 over F_2

Abstract

A projective, smooth, absolutely irreducible algebraic curve X of genus g defined over a finite field F_q is called optimal if for every other such genus g curve Y over F_q one has $\#Y(F_q)\le \#X(F_q)$. In this paper we show that for $g\le 5$ there is a unique optimal genus g curve over F_2. For g=6 there are precisely two and for g=7 there are at least two.