Disjoint weighing matrices
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Disjoint weighing matrices Hadi Kharaghani1
· Sho Suda2 · Behruz Tayfeh-Rezaie3
Received: 6 February 2020 / Accepted: 3 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The notion of disjoint weighing matrices is introduced as a generalization of orthogonal designs. A recursive construction along with a computer search leads to some infinite classes of disjoint weighing matrices, which in turn are shown to form commutative association schemes with 3 or 4 classes. Keywords Weighing matrix · Orthogonal design · Disjoint skew weighing matrices · Association scheme · circulant matrix
1 Introduction A weighing matrix of order n and weight w, denoted W (n, w), is an n × n (0, 1, −1)matrix W such that W W = w In , where In denotes the identity matrix of order n. An orthogonal design of order n and type (w1 , . . . , wk ) in variables x1 , . . . , xk , denoted by O D(n; w1 , . . . , wk ), is a square (0, ±x1 , . . . , ±xk )-matrix D, where x1 , . . . , xk are distinct commuting indeterminates, such that D D = (w1 x12 + · · · + wk xk2 )In .
Dedicated to professor Arasu on the occasion of his 65th birthday.
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Hadi Kharaghani [email protected] Sho Suda [email protected] Behruz Tayfeh-Rezaie [email protected]
1
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
2
Department of Mathematics, National Defense Academy of Japan, 2-10-20 Hashirimizu, Yokosuka, Kanagawa 239-8686, Japan
3
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
123
Journal of Algebraic Combinatorics
k Splitting the matrix D = i=1 xi Wi , where each Wi is a weighing matrix W (n, wi ), k then i=1 Wi is a (0, 1, −1)-matrix. Furthermore, the weighing matrices Wi satisfy the antiamicability condition Wi W j + W j Wi = On for any distinct pair i, j, where On denotes the zero matrix of order n. The antiamicability condition imposes restrictions on the number of disjoint weighing matrices stemming from orthogonal designs. Relaxing this condition provides a much larger class of disjoint weighing matrices enjoying very nice properties. This has motivated us to study these matrices. For a (0, 1, −1)-matrix X , denote by |X | the matrix obtained by replacing −1 with 1 in X . Definition 1.1 Let n, k, w1 , . . . , wk be positive integers and let Wi be aweighing k |Wi | matrix of order n and weight wi for i ∈ {1, . . . , k}. If Wi are disjoint, that is i=1 is a (0, 1)-matrix, then we call W1 , . . . , Wk disjoint weighing matrices denoting it by DW (n; w1 , . . . , wk ). The variety of applications of disjoint weighing matrices includes their use in the construction of Bush-type Hadamard matrices [8], their role as the building blocks of orthogonal designs [4] and their use in the construction of symmetric group divisible designs and association schemes [10]. Craigen [2] used these matrices in his study of disjoint weighing designs. k The notation DW(n; [ n−1 k ] ) will be used to show k dis
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