Abstract
It is well-known that a complex Hadamard matrix of order \(2n\) can be used to construct a Hadamard matrix of order \(4n\). Let \(\gamma \) be a primitive complex cube root of unity. In this paper, we describe a method for obtaining a Hadamard matrix of order \(4n\) from a Butson-type generalized Hadamard matrix \(BH(n,6)\) whose entries are drawn from the set \(\{\pm \gamma ,\pm \gamma ^2\}\) of non-real complex sixth roots of unity. We denote such a matrix by \(BH(n,6)\). No \(BH(n,6)\) can exist for odd order \(n\) whose squarefree part is divisible by a prime \(p\equiv 2\pmod 3\). We exhibit examples of such “unreal” \(BH(n,6)\) for all orders \(n<19\) not ruled out by the above condition. We obtain unreal \(BH(n,6)\)’s for \(n=3^{k}hm\), where \(k\) is a nonnegative integer, \(h\) is the order of a Hadamard matrix, and \(m \in \{1,7,10,13\}\).
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Acknowledgments
We acknowledge numerous helpful observations by the referees and computations by the second author’s student Ivan Livinskyi, which led to improved articulation concerning equivalence classes. Although completed in 2008, this work was held back after acceptance in 2009 for minor technical reasons. Then the project was neglected for a period while our lives were complicated by cancer. In summer 2010 the second author’s wife passed away. Shortly thereafter the third author, Warwick de Launey, also succumbed to the disease, which he had battled for several years. The remaining two authors would like to dedicate this paper to Warwick’s unquenchable spirit and creative insights, many examples of which are found herein. This work was supported by NSERC Grant.
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Communicated by A. Winterhof.
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Compton, B., Craigen, R. & de Launey, W. Unreal \(BH(n,6)\)’s and Hadamard matrices. Des. Codes Cryptogr. 79, 219–229 (2016). https://doi.org/10.1007/s10623-015-0045-y
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DOI: https://doi.org/10.1007/s10623-015-0045-y