Abstract
A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ≠ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.
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Acknowledgements
This work was inspired by the discussion after Will Orrick’s talk at the ADTHM’14 workshop, and much of the work was undertaken at the workshop. The authors are grateful to the workshop organisers and to BIRS. Research supported by ARC grants FT110100065 and DP120103067.
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Dedicated to Hadi Kharaghani on the occasion on his 70th birthday
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Catháin, P.Ó., Wanless, I.M. (2015). Trades in Complex Hadamard Matrices. 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_18
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DOI: https://doi.org/10.1007/978-3-319-17729-8_18
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17728-1
Online ISBN: 978-3-319-17729-8
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