On sizes of complete arcs in PG(2,q)

2010/11/15 by Daniele Bartoli, Alexander A. Davydov, Bartoli, Daniele +7 · 1 citation
Computer Science · Mathematics · #51E21 #51E22 (Primary) #94B05 (Secondary) #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #cs.IT #math.CO #math.IT #msc:51E21 #msc:51E22 #msc:94B05

paper · pdf · doi:10.48550/arxiv.1011.3347

27 pages, 4 figures, 5 tables

arxiv created 2011/05/23 · arxiv updated 2011/05/24

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 <= 4561 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, T2=4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192. From these new bounds it follows that for q <= 2593 and q=2693,2753, the relation t₂(2,q) < 4.5√(q) holds. Also, for q <= 4561 we have t₂(2,q) < 4.75√(q). It is showed that for 23 <= q <= 4561 and q∈ T2∪ 2¹⁴,2¹⁵,2¹⁸, the inequality t₂(2,q) < √(q)ln^0.75q 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 k_min <= k <= 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 <= 1367 and 2003 <= q <= 2063. There is reason to suppose that the arcs are complete for all q > 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.

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• Upper bounds on the smallest size of a complete arc in the plane PG(2,q)

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