New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

Abstract

In the projective planes PG(2, q), more than 1230 new small complete arcs are obtained for ${q \leq 13627}$ and ${q \in G}$ where G is a set of 38 values in the range 13687,..., 45893; also, ${2^{18} \in G}$. This implies new upper bounds on the smallest size t ₂(2, q) of a complete arc in PG(2, q). From the new bounds it follows that

$$t_{2}(2, q) < 4.5\sqrt{q} \, {\rm for} \, q \leq 2647$$

and q = 2659,2663,2683,2693,2753,2801. Also,

$$t_{2}(2, q) < 4.8\sqrt{q} \, {\rm for} \, q \leq 5419$$

and q = 5441,5443,5449,5471,5477,5479,5483,5501,5521. Moreover,

$$t_{2}(2, q) < 5\sqrt{q} \, {\rm for} \, q \leq 9497$$

and q = 9539,9587,9613,9623,9649,9689,9923,9973. Finally,

$$t_{2}(2, q) <5 .15\sqrt{q} \, {\rm for} \, q \leq 13627$$

and q = 13687,13697,13711,14009. Using the new arcs it is shown that

$$t_{2}(2, q) < \sqrt{q}\ln^{0.73}q {\rm for} 109 \leq q \leq 13627\, {\rm and}\, q \in G.$$

Also, as q grows, the positive difference ${\sqrt{q}\ln^{0.73} q-\overline{t}_{2}(2, q)}$ has a tendency to increase whereas the ratio ${\overline{t}_{2}(2, q)/(\sqrt{q}\ln^{0.73} q)}$ tends to decrease. Here ${\overline{t}_{2}(2, q)}$ is the smallest known size of a complete arc in PG(2,q). These properties allow us to conjecture that the estimate ${t_{2}(2,q) < \sqrt{q}\ln ^{0.73}q}$ holds for all ${q \geq 109.}$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms. Finally, new forms of the upper bound on t ₂(2,q) are proposed.