[Paper Review] Upper bounds on the smallest size of a complete arc in the plane PG(2,q)

Daniele Bartoli, Alexander A. Davydov+3 more|arXiv (Cornell University)|Nov 15, 2011

Finite Group Theory Research15 references3 citations

TL;DR

This paper presents new upper bounds on the smallest size of a complete arc in the projective plane $PG(2,q)$, using computer-aided randomized greedy algorithms to construct smaller complete arcs. It proves that $t_2(2,q) < 4.5\sqrt{q}$ for $q \leq 2621$ and $q=2659,\dots,2753$, $t_2(2,q) < 4.8\sqrt{q}$ for $q \leq 5399$ and certain primes, and $t_2(2,q) < 5\sqrt{q}$ for $q \leq 9067$, supporting the conjecture that $t_2(2,q) < \sqrt{q}\ln^{0.75}q$ for $q \geq 23$. These bounds are derived from extensive computational searches and statistical analysis of arc size distributions.

ABSTRACT

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for q &lt;= 9109. From these new bounds it follows that for q &lt;= 2621 and q = 2659,2663,2683,2693,2753,2801, the relation t_{2}(2,q) &lt; 4.5\sqrt{q} holds. Also, for q &lt;= 5399 and q = 5413,5417,5419,5441,5443,5471,5483,5501,5521, we have t_{2}(2,q) &lt; 4.8\sqrt{q}. Finally, for q &lt;= 9067 it holds that t_{2}(2,q) &lt; 5\sqrt{q}. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms.

Motivation & Objective

• To improve upper bounds on the smallest size $t_2(2,q)$ of a complete arc in the projective plane $PG(2,q)$.
• To extend and refine prior conjectures on the asymptotic growth of $t_2(2,q)$, particularly $t_2(2,q) < \sqrt{q}\ln^{0.75}q$ for $q \geq 23$.
• To provide computationally verified bounds for $t_2(2,q)$ up to $q = 9109$ using randomized greedy algorithms.
• To analyze the distribution of arc sizes and assess the accuracy of predicted bounds using statistical measures like $\overline{D}_q(0.75)$.
• To support the conjecture that $t_2(2,q) < 5\sqrt{q}$ for all $q \leq 8192$, which is proven in this work.

Proposed method

• A computer-aided search using randomized greedy algorithms to construct small complete arcs in $PG(2,q)$ for $q \leq 9109$.
• Computation of the average density $\overline{D}_q(0.75)$ of complete arcs relative to $\sqrt{q}\ln^{0.75}q$ to assess bound quality.
• Definition of $\widehat{t}_2(2,q) = \overline{D}_{\text{aver}}(0.75,173)\sqrt{q}\ln^{0.75}q$ as a predicted upper bound for $t_2(2,q)$.
• Calculation of the deviation $\overline{\Delta}_q = \overline{t}_2(2,q) - \widehat{t}_2(2,q)$ and percentage error $\overline{P}_q$ to evaluate bound tightness.
• Statistical analysis of $\overline{D}_q(0.75)$ across intervals of $q$ to validate the robustness of the $\sqrt{q}\ln^{0.75}q$ bound.
• Use of previously known bounds and constructions as baselines, including $4\sqrt{q}$, $4.5\sqrt{q}$, and $4.8\sqrt{q}$, to contextualize new results.

Experimental results

Research questions

• RQ1 Does $t_2(2,q) < 4.5\sqrt{q}$ hold for all $q \leq 2621$ and specific primes in that range?
• RQ2 Can the bound $t_2(2,q) < 4.8\sqrt{q}$ be extended to $q \leq 5399$ and additional primes?
• RQ3 Is the conjecture $t_2(2,q) < 5\sqrt{q}$ true for all $q \leq 8192$, as proven in this work?
• RQ4 How close is the current smallest known arc size $\overline{t}_2(2,q)$ to the predicted bound $\sqrt{q}\ln^{0.75}q$?
• RQ5 Can the statistical measure $\overline{D}_q(0.75)$ be used to reliably estimate the tightness of asymptotic upper bounds on $t_2(2,q)$?

Key findings

• For $q \leq 2621$ and $q = 2659, 2663, 2683, 2693, 2753$, it holds that $t_2(2,q) < 4.5\sqrt{q}$.
• For $q \leq 5399$ and $q = 5413, 5417, 5419, 5441, 5443, 5471, 5483, 5501, 5521$, $t_2(2,q) < 4.8\sqrt{q}$.
• For $q \leq 9067$, the bound $t_2(2,q) < 5\sqrt{q}$ is confirmed, proving Conjecture 1.2 from prior work.
• The average deviation $\overline{\Delta}_q$ between the smallest known arc size and the predicted $\sqrt{q}\ln^{0.75}q$ bound is bounded by $-3.70 < \overline{\Delta}_q < 0.81$ for $173 \leq q \leq 9109$.
• The percentage error $\overline{P}_q$ is bounded by $-0.94\% < \overline{P}_q < 0.79\%$ for $q < 1000$, decreasing with increasing $q$, indicating tighter agreement as $q$ grows.
• The statistical analysis of $\overline{D}_q(0.75)$ shows consistent values between 0.947 and 0.9634 across intervals, supporting the $\sqrt{q}\ln^{0.75}q$ bound as a reliable upper estimate.