tailieunhanh - Báo cáo toán học: "On the twin designs with the Ionin–type parameters"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On the twin designs with the Ionin–type parameters. | On the twin designs with the Ionin-type parameters H. Kharaghani Department of Mathematics Computer Science University of Lethbridge Lethbridge Alberta T1K 3M4 Canada hadi@ Submitted July 31 1999 Accepted October 20 1999 Dedicated to Professor Reza Khosrovshahi on the occasion of his 60th birthday Keywords Symmetric design regular Hadamard matrix Bush-type Hadamard matrix design with Ionin-type parameters balanced generalized weighing matrix weighing matrix. MR Subject Code 05B05 Abstract Let 4n2 be the order of a Bush-type Hadamard matrix with q 2n 1 2 a prime power. It is shown that there is a weighing matrix W 4 qm qm-1 q 1 n2 4qmn2 which includes two symmetric designs with the Ionin-type parameters V 4 qm qm q 1 n2 K qm 2n2 n X qm n2 n for every positive integer m. Noting that Bush-type Hadamard matrices of order 16n2 exist for all n for which an Hadamard matrix of order 4n exist this provides a new class of symmetric designs. Thanks to Stephen Ney for proving me wrong on my first choice of the cyclic group by writing a program and applying the group. Wolfgang Holzmann as always was a great helper. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R1 2 1 Introduction Recently Ionin 3 introduced an elegant method to use a very special class of regular Hadamard matrix of order 36 in a class of balanced generalized weighing matrices to construct a large class of symmetric designs. The key to his construction is the existence of a class of balanced generalized weighing matrices BGW qm qm-1 q 1 qm qm qm 1 over a cyclic group of order t where q is a prime power m is a positive integer and t is a divisor of q 1. A balanced generalized weighing matrix BGW V K X over a group G is a matrix W Wj of order V with wij 2 G u 0 such that each row and column of W has K non-zero entries and for each k l the multiset wkjw-j1 1 j v Wkj 0 wlj 0 contains X G copies of every element of G. For his construction Ionin starts with what he calls a starting regular Hadamard .