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Báo cáo toán học: "The Existence of FGDRP(3, g u)′s"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: The Existence of FGDRP(3, g u)′s. | The Existence of FGDRP 3 gu s Jie Yan and Chengmin Wang School of Science Jiangnan University Wuxi 214122 China jyan7906@yahoo.com.cn wcm@jiangnan.edu.cn Submitted Sep 9 2008 Accepted Mar 3 2009 Published Mar 13 2009 Mathematics Subject Classification 05B05 Abstract By an FGDRP 3 gu we mean a uniform frame X G A of block size 3 index 2 and type gu where the blocks of A can be arranged into a gu 3 X gu array. This array has the properties 1 the main diagonal consists of u empty subarrays of sizes g 3 X g 2 the blocks in each column form a partial parallel class partitioning X G for some G E G while the blocks in each row contain every element of X G 3 times and no element of G for some G E G. The obvious necessary conditions for the existence of an FGDRP 3 gu are u 5 and g 0 mod 3 . In this paper we show that these conditions are also sufficient with the possible exceptions of g u E 6 15 9 18 9 28 9 34 30 15 . 1 Introduction In this paper we use 1 and 2 as our standard design-theoretic references. A group divisible design or a K A -GDD in short is a triple X G A where X is a finite set of v points G G0 G1 Gu-1 is a partition of X into u subsets called groups and A is a collection of subsets called blocks of X with A E K for any A E A such that every pair of points from distinct groups occurs in exactly A blocks and no pair of points from the same group occurs in any block. The group type or the type of a K A -GDD is the multiset T G0 G1 Gu-1 which is often described by an exponential notation. When K consists of a single number k the notation k A -GDD is used. Further we denote k 1 -GDD as k-GDD. A K A -GDD of type 1v is known as a pairwise balanced design PBD or a v K A -PBD. In this case the group set G is the same as the point set and hence the symbol G is often omitted from the notation X G A . Remark that a transversal design TD or a TD k n is defined as a k-GDD of type nk. Research is supported by the Natural Science Foundation of China under Grant No. .