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Báo cáo toán học: "New symmetric designs from regular Hadamard matrices"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: New symmetric designs from regular Hadamard matrices. | New symmetric designs from regular Hadamard matrices Yury J. Ionin Department of Mathematics Central Michigan University Mt. Pleasant Mi 48859 USA yury.ionin@cmich.edu November 29 1997 Abstract For every positive integer m we construct a symmetric y k A -design with parameters y h 2h- m-1 k h 2h - 1 2m-1 and A h h - 1 2h - 1 2m-2 where h 3 2d and 2h 1 is a prime power. For m 2 and d 1 these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order 9 4d. Submitted October 30 1997 Accepted November 17 1997 MR Subject Number 05B05 Keywords Symmetric design regular Hadamard matrix balanced generalized weighing matrix 1 Introduction Let v k A 0be integers. A symmetric v k A -design is an incidence structure P B where P is a set of cardinality v the point-set and B is a family of v k-subsets blocks of P such that any two distinct points are contained in exactly A blocks. If P p1 . pv and B B1 . Bv then the 0 1 -matrix M mij of order v where mij 1 if and only if Pj 2 Bi is the incidence matrix of the design. A 0 1 -matrix X of order v is the incidence matrix of a symmetric v k A -design if and only if it satisfies the equation XXT k A I AJ where I is the identity matrix and J is the all-one matrix of order v. For references see 1 or 3 Chapter 5 . A Hadamard matrix of order n is an n by n matrix H with entries equal to 1 satisfying HHT nl. A Hadamard matrix is regular if its row and column sums are constant. This sum is always even and if we denote it 2h then the order of the matrix is equal to 4h2. Replacing 1s in a regular Hadamard matrix of order 4h2 by 0s yields the incidence matrix of a symmetric 4h2 2h2 h h2 h -design usually The author acknowledges with thanks the Central Michigan University Research Professor award. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1997 R1 2 called a Menon design. Conversely replacing 0s by 1s in the incidence matrix of a symmetric 4h2