Mathematics – Combinatorics
Scientific paper
2010-11-15
Mathematics
Combinatorics
27 pages, 4 figures, 5 tables
Scientific paper
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 <= q <= 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 <= 2593 and q=2693,2753, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 4561 we have t_{2}(2,q) < 4.75\sqrt{q}. It is showed that for 23 <= q <= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) < \sqrt{q}ln^{0.75}q 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.
Bartoli Daniele
Davydov Alexander A.
Faina Giorgio
Marcugini Stefano
Pambianco Fernanda
No associations
LandOfFree
On sizes of complete arcs in PG(2,q) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On sizes of complete arcs in PG(2,q), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On sizes of complete arcs in PG(2,q) will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-298633