[Paper Review] On sizes of complete arcs in PG(2,q)

Daniele Bartoli, Alexander A. Davydov+3 more|arXiv (Cornell University)|Nov 15, 2010

Finite Group Theory Research7 references3 citations

TL;DR

This paper presents new upper bounds on the smallest size of a complete arc in the projective plane $PG(2,q)$ for $q \leq 4561$, using computer-aided randomized greedy algorithms and novel constructive methods. It establishes that $t_2(2,q) < 4.5\sqrt{q}$ for $q \leq 2593$ and $t_2(2,q) < 4.75\sqrt{q}$ for $q \leq 4561$, and conjectures $t_2(2,q) < \sqrt{q} \ln^{0.75}q$ holds for all $q \geq 23$, with complete arcs constructed for $k$-sizes in large intervals around $q/3$ to $q/2$. The results significantly extend known bounds and the spectrum of complete arc sizes.

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 853 &lt;= q &lt;= 4561 and q\in T1\cup T2 where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457, 461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631, 641,661,673,677,683,691, 709}, T2={4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192}. From these new bounds it follows that for q &lt;= 2593 and q=2693,2753, the relation t_{2}(2,q) &lt; 4.5\sqrt{q} holds. Also, for q &lt;= 4561 we have t_{2}(2,q) &lt; 4.75\sqrt{q}. It is showed that for 23 &lt;= q &lt;= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) &lt; \sqrt{q}ln^{0.75}q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q &gt;= 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} &lt;= k &lt;= k_{max} where k_{min} is of order q/3 or q/4 while k_{max} has order q/2. The completeness of the arcs obtained by the new constructions is proved for q &lt;= 1367 and 2003 &lt;= q &lt;= 2063. There is reason to suppose that the arcs are complete for all q &gt; 1367. New sizes of complete arcs in PG(2,q) are presented for 169 &lt;= q &lt;= 349 and q=1013,2003.

Motivation & Objective

• To improve upper bounds on the smallest size $t_2(2,q)$ of a complete arc in $PG(2,q)$ for large $q$.
• To extend the known spectrum of possible sizes of complete arcs in $PG(2,q)$, particularly for $q \leq 4561$.
• To develop and validate new constructive methods for generating complete arcs with controlled sizes.
• To conjecture a universal asymptotic upper bound $t_2(2,q) < \sqrt{q} \ln^{0.75}q$ for all $q \geq 23$.

Proposed method

• Computer search using randomized greedy algorithms to discover new small complete arcs for $853 \leq q \leq 4561$ and specific $q \in T_1 \cup T_2$.
• Proposing three new constructive families of $k$-arcs in $PG(2,q)$ that generate complete arcs for $k$ in intervals from $\sim q/3$ to $\sim q/2$.
• Using conic and cubic curve points as starting sets in greedy algorithms to efficiently explore arc size spectra.
• Proving completeness of constructed arcs for $q \leq 1367$ and $2003 \leq q \leq 2063$, with strong evidence for completeness in larger $q$.
• Leveraging known geometric constructions (e.g., Segre's idea) by selecting points from conics or cubics to guide arc generation.
• Applying theoretical bounds and corollaries (e.g., Corollary 5.34, Theorem 5.35) to support conjectures on asymptotic behavior.

Experimental results

Research questions

• RQ1 What are the tightest known upper bounds on $t_2(2,q)$ for $q \leq 4561$, particularly for prime powers not previously covered?
• RQ2 Can new constructive methods generate complete arcs with $k$-sizes spanning a large interval from $\sim q/3$ to $\sim q/2$?
• RQ3 Does the bound $t_2(2,q) < \sqrt{q} \ln^{0.75}q$ hold for all $q \geq 23$, and can it be proven or strongly supported?
• RQ4 What is the extent of the spectrum of complete arc sizes in $PG(2,q)$ for $q \leq 4561$, and can all sizes from $t_2(2,q)$ to $M_q$ be realized?
• RQ5 How effective are randomized greedy algorithms combined with conic-based initialization in discovering small complete arcs?

Key findings

• For $q \leq 2593$ and $q = 2693, 2753$, it holds that $t_2(2,q) < 4.5\sqrt{q}$, improving prior bounds.
• For $q \leq 4561$, the bound $t_2(2,q) < 4.75\sqrt{q}$ is established, extending known results.
• For $23 \leq q \leq 4561$ and $q \in T_2 \cup \{2^{14}, 2^{15}, 2^{18}\}$, the inequality $t_2(2,q) < \sqrt{q} \ln^{0.75}q$ is proven.
• The authors conjecture that $t_2(2,q) < \sqrt{q} \ln^{0.75}q$ holds for all $q \geq 23$, based on extensive computational evidence.
• New constructions generate complete $k$-arcs for all $k$ in intervals $k_{\min} \leq k \leq k_{\max}$ with $k_{\min} \sim q/3$ or $q/4$ and $k_{\max} \sim q/2$, verified for $q \leq 1367$ and $2003 \leq q \leq 2063$.
• For $169 \leq q \leq 349$ and $q = 1013, 2003$, complete $k$-arcs exist for all sizes from $\overline{t}_2(2,q)$ to $M_q$, as confirmed by combined constructions and greedy search.