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Yet More Projective Curves Over F2

Chris Lomont

TL;DR

This work tackles the problem of finding plane curves over $\mathbb{F}_2$ of degree $<7$ whose smooth models have many rational points over extensions $\mathbb{F}_{2^m}$. It combines genus bounds, singularity analysis, Ragot-based irreducibility tests, and heavy computational pruning to exhaustively search and count $\mathbb{F}_q$-rational points on desingularized curves, across $q=2^m$ with $m\in\{3,...,11\}$. The authors report numerous new curves that meet or nearly achieve Serre, Ihara, and Lauter bounds, providing explicit equations and compiling best-known $N_q(g)$ values and thereby advancing the known landscape of maximal rational points on curves over small fields. The results have practical relevance for constructing algebraic-geometric codes from desingularized curves, and the work demonstrates the viability of large-scale searches for high-point curves on both supercomputers and high-end desktops.

Abstract

All binary plane curves of degree less than 7 are examined for curves with a large number of Fq rational points on their smooth model, for q = 2^m ; m = 3, 4,...,11. Previous results are improved, and many new curves are found meeting or close to Serre's, Lauter's, and Ihara's upper bounds.

Yet More Projective Curves Over F2

TL;DR

This work tackles the problem of finding plane curves over of degree whose smooth models have many rational points over extensions . It combines genus bounds, singularity analysis, Ragot-based irreducibility tests, and heavy computational pruning to exhaustively search and count -rational points on desingularized curves, across with . The authors report numerous new curves that meet or nearly achieve Serre, Ihara, and Lauter bounds, providing explicit equations and compiling best-known values and thereby advancing the known landscape of maximal rational points on curves over small fields. The results have practical relevance for constructing algebraic-geometric codes from desingularized curves, and the work demonstrates the viability of large-scale searches for high-point curves on both supercomputers and high-end desktops.

Abstract

All binary plane curves of degree less than 7 are examined for curves with a large number of Fq rational points on their smooth model, for q = 2^m ; m = 3, 4,...,11. Previous results are improved, and many new curves are found meeting or close to Serre's, Lauter's, and Ihara's upper bounds.

Paper Structure

This paper contains 15 sections, 2 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Computation
  3. Computational Results
  4. $\mathbb{F}_{8}$
  5. $\mathbb{F}_{16}$
  6. $\mathbb{F}_{32}$
  7. $\mathbb{F}_{64}$
  8. $\mathbb{F}_{128}$
  9. $\mathbb{F}_{256}$
  10. $\mathbb{F}_{512}$
  11. $\mathbb{F}_{1024}$
  12. $\mathbb{F}_{2048}$
  13. Tallies
  14. Conclusions
  15. Comments

Key Result

Theorem 1

Let $C\subseteq \mathbb{P}^2$ be a plane curve of degree d with singularities $P_1, P_2, ...,P_r$, with multiplicities $m_1,m_2,...,m_r$, for $r\geq 2$. Then $\sum^r_{i=1}m_i \leq \lfloor \frac{d}{2} \rfloor r + 1$ if d is odd, and $\sum^r_{i=1}m_i \leq \lfloor \frac{d}{2} \rfloor r$ if d is even.

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof