Abstract
Craigen introduced and studied signed group Hadamard matrices extensively following Craigen’s lead, studied and provided a better estimate for the asymptotic existence of signed group Hadamard matrices and consequently improved the asymptotic existence of Hadamard matrices. In this paper, we introduce and study signed group orthogonal designs (SODs). The main results include a method for finding SODs for any k-tuple of positive integer and then an application to obtain orthogonal designs from SODs, namely, for any k-tuple \(\big(u_{1},u_{2},\ldots,u_{k}\big)\) of positive integers, we show that there is an integer \(N = N(u_{1},u_{2},\ldots,u_{k})\) such that for each n ≥ N, a full orthogonal design (no zero entries) of type \(\big(2^{n}u_{1},2^{n}u_{2},\ldots,2^{n}u_{k}\big)\) exists.
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Acknowledgements
The paper constitutes a part of the author’s Ph.D. thesis written under the direction of Professor Hadi Kharaghani at the University of Lethbridge. The author would like to thank Professor Hadi Kharaghani for introducing the problem and his very useful guidance toward solving the problem and also Professor Rob Craigen for his time and great help.
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Dedicated to Hadi Kharaghani on the occasion on his 70th birthday
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Ghaderpour, E. (2015). Signed Group Orthogonal Designs and Their Applications. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_9
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DOI: https://doi.org/10.1007/978-3-319-17729-8_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17728-1
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