tailieunhanh - Báo cáo toán học: "On small dense sets in Galois planes"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On small dense sets in Galois planes. | On small dense sets in Galois planes M. Giulietti Dipartimento di Matematica e Informatica Università di Perugia Italy giuliet@ Submitted Jul 17 2007 Accepted Oct 31 2007 Published Nov 5 2007 Mathematics Subject Classification 51E20 Abstract This paper deals with new infinite families of small dense sets in desarguesian projective planes PG 2 q . A general construction of dense sets of size about 3q2 3 is presented. Better results are obtained for specific values of q. In several cases an improvement on the best known upper bound on the size of the smallest dense set in PG 2 q is obtained. 1 Introduction A dense set K in PG 2 q the projective plane coordinatized over the finite field with q elements Fq is a point-set whose secants cover PG 2 q that is any point of PG 2 q belongs to a line joining two distinct points of K. As well as being a natural geometrical problem the construction of small dense sets in PG 2 q is relevant in other areas of Combinatorics as dense sets are related to covering codes see Section 4 and defining sets of block designs see 2 also it has been recently pointed out in 13 that small dense sets are connected to the degree diameter problem in Graph Theory 17 . A straightforward counting argument shows that a trivial lower bound for the size k of a dense set in PG 2 q is k p2q see . 19 . On the other hand for q square there is a nice example of a dense set of size 3pq namely the union of three non-concurrent lines of a subplane of PG 2 q of order pq. If q is not a square however the trivial lower bound is far away from the size of the known examples. The existence of dense sets of size LWqlogqj was shown by means of probabilistic methods see 2 14 . The smallest dense sets explicitly constructed so far have size approximately cq4 with c a constant independent on q see 1 9 18 for This research was performed within the activity of GNSAGA of the Italian INDAM with the financial support of the Italian Ministry MIUR project .