Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Some Pairwise Balanced Designs. | Some Pairwise Balanced Designs Malcolm Greig Greig Consulting 207-170 East Fifth St. North Vancouver BC Canada V7L 4L4 greig@ Submitted Oct. 6 1998 Accepted Oct. 23 1999. Abstract A pairwise balanced design B K v is a block design on v points with block sizes taken from K and with every pair of points occurring in a unique block for a fixed K B K is the set of all v for which a B K v exists. A set S is a PBD-basis for the set T if T B S . Let Na m n n a mod mg and N m n n mg with Q the corresponding restriction of N to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that Q1 8 is a basis for N1 8 E where E is a set of 5 definite and 117 possible exceptions. 2. We construct a 78 element basis for N1 8 with at most 64 inessential elements. 3. Bennett and Zhu have shown that Q 8 is a basis for N 8 E where E0 is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. Constructions for 35 of these possible exceptions are known. Finally we provide brief details of some improvements and corrections to the generating exception sets published in The CRC Handbook of Combinatorial Designs. Key words and phrases BIBD Pairwise Balanced Design AMS subject classifications Primary 05B05. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R13 2 1 Introduction A pairwise balanced design B K v is a block design on v points with block sizes taken from K and with every pair of points occurring in a unique block for a fixed K B K is the set of all v for which a B K v exists. A set S is a PBD-basis for the set T if T B S . Let Na m n n a mod mg and N m n n mg with Q the corresponding restriction of N to prime powers. This paper addresses the existence of three PBD-basis sets. The opening sections deal with useful known results and more general constructions. In Section 5 we give constructions that are particularly useful for the first of our two problems then in Section 6 we show .
