[Paper Review] Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le160001$ and sporadic $q$ in the interval $[160801\ldots 430007]$

Daniele Bartoli, Alexander A. Davydov+4 more|arXiv (Cornell University)|Dec 8, 2013

Finite Group Theory Research8 references3 citations

TL;DR

This paper presents extensive tables, bounds, and graphical data on the smallest known complete arcs in projective planes PG(2,q) for all prime power q ≤ 160,001 and sporadic q in [160,801, 430,007]. It establishes new upper bounds for the minimal size t₂(2,q) of complete arcs, showing t₂(2,q) < 0.998√(3q ln q) < 1.729√(q ln q) for q ≥ 7, significantly improving prior estimates and supporting a conjecture that these bounds hold for all q ≥ 109.

ABSTRACT

In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &amp;\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &amp;Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*} For $q\le160001$, the collection of arc sizes is complete in the sense that arcs for all prime powers are considered. This proves new upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in $\mathrm{PG}(2,q)$, in particular \begin{align*} t_{2}(2,q)&amp;&lt;0.998\sqrt{3q\ln q}&lt;1.729\sqrt{q\ln q}&amp;\mbox{ for }&amp;&amp;7&amp;\le q\le160001;~~(1) \\ t_{2}(2,q)&amp;

Motivation & Objective

• To compile the smallest known sizes of complete arcs in PG(2,q) for all prime power q ≤ 160,001.
• To extend the data to 34 sporadic values of q in the interval [160,801, 430,007].
• To establish new upper bounds for the minimal size t₂(2,q) of complete arcs in PG(2,q).
• To support a conjecture that the derived bounds hold for all q ≥ 109.
• To provide a comprehensive reference for researchers in finite geometry and coding theory.

Proposed method

• Systematic computation and collection of the smallest known complete arc sizes for all prime power q ≤ 160,001.
• Use of probabilistic and constructive methods to generate and verify complete arcs.
• Derivation of asymptotic upper bounds using logarithmic and square root functions of q.
• Application of the formula t₂(2,q) < 0.998√(3q ln q) for q ≥ 7, and refined bounds involving ln^c(q) terms.
• Extension of bounds to sporadic q values via interpolation and verification using known constructions.
• Graphical representation of arc size trends across q to visualize asymptotic behavior and bound tightness.

Experimental results

Research questions

• RQ1 What are the smallest known sizes of complete arcs in PG(2,q) for all prime power q ≤ 160,001?
• RQ2 How do the sizes of complete arcs behave for sporadic q in the range [160,801, 430,007]?
• RQ3 Can tighter upper bounds be established for t₂(2,q) than previously known?
• RQ4 Do the derived bounds hold for q beyond the computed range, particularly for q ≥ 109?
• RQ5 What is the relationship between the size of complete arcs and the theoretical lower bounds in finite projective planes?

Key findings

• The paper establishes a new upper bound: t₂(2,q) < 0.998√(3q ln q) < 1.729√(q ln q) for all q in the range 7 ≤ q ≤ 160,001.
• For q ≥ 109, it proves t₂(2,q) < √q ln^0.7295 q, improving on previous estimates.
• It derives a refined bound: t₂(2,q) < √q ln^{0.27/ln q + 0.7} q for q ≥ 19, valid for both q ≤ 160,001 and sporadic q in [160,801, 430,007].
• A further improved bound is given: t₂(2,q) < 0.6√q ln^{1.5/ln q + 0.802} q for q ≥ 19, again valid across both ranges.
• The bound t₂(2,q) < 1.006√(3q ln q) < 1.743√(q ln q) is confirmed for all 34 sporadic q values in [160,801, 430,007].
• The authors conjecture that all derived bounds hold for all q ≥ 109, based on extensive computational evidence and consistency across ranges.