COMBINATORIAL PROBLEMS FOR ABELIAN GROUPS ARISING FROM GEOMETRY

This paper deals with elementary problems on complexes of abelian groups related to finite geometry, in particular to arcs and blocking sets of finite projective planes. Arcs contained in cubic curves led us to the notion of a 3-independent subset in abelian groups. Various examples of complete arcs containing only three points outside a conic were constructed by KORCHMÁROS [6) using 2 -(m, n) isolated sets. In this paper we survey the known results and constructions concerning 3-independent and 2 (m, n) isolated sets. Moreover we obtain some new bounds for their size and give some new examples showing that the lower and upper bounds are sharp regarding their order of magnitude. Finally, we will show how the methods and constructions of the previous sections can be applied to the problem of blocking sets contained in the union of three lines and answer a question of CAMERON [1]

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

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.Comment: 27 pages, 4 figures, 5 table