Flowserve Corp, USA

Abstract

New upper bounds on the smallest size t2(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853≤q≤5107 and q∈T1∪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}, and T2={5119,5147,5153,5209,5231,5237,5261,5279,5281,5303,5347,5641,5843,6011,8192}. From these new bounds it follows that for q≤2593 and q=2693,2753, the relation t2(2,q)<4.5q holds. Also, for q≤5107 we have t2(2,q)<4.79q. It is shown that for 23≤q≤5107 and q∈T2∪{214,215,218}, the inequality t2(2,q)<qln0.75q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q≥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 kmin≤k≤kmax, where kmin is of order 13q or 14q while kmax has order 12q. The completeness of the arcs obtained by the new constructions is proved for q≤2063. There is reason to suppose that the arcs are complete for all q>2063. New sizes of complete arcs in PG(2,q) are presented for 169≤q≤349 and q=1013,2003.