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Exotic complex Hadamard matrices and their equivalence

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Abstract

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe and Jones (C R Acad Sci Paris 311(Série I): 147–150, 1990), and Munemasa and Watatani (C R Acad Sci Paris 314(Série I): 329–331, 1992) and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n − 8)×(n − 8) minors of real Hadamard matrices (Koukouvinos et al., Linear Algebra Appl 371:111–124, 2003).

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Notes

  1. This problem and its elementary solution was presented by Jennifer Seberry in the International Conference on Design Theory and Applications in Galway, 2009.

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Acknowledgements

The author is greatly indebted to the referees for their valuable comments which helped to improve this manuscript. After the first version of the paper was circulated Professor Chris Godsil pointed out that equation (2) implies v ≤ 4(k − λ). By combining this with the well-known lower bound 4(k − λ) − 1 ≤ v one can see that induced complex Hadamard matrices correspond to the Hadamard- and Menon designs only. We acknowledge his kind contribution.

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  1. Department of Mathematics and its Applications, Central European University, 1051, Nádor u. 9, Budapest, Hungary

    Ferenc Szöllősi

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  1. Ferenc Szöllősi
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Correspondence to Ferenc Szöllősi.

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Dedicated to Professor Warwick de Launey on the occasion of his 50th birthday.

This work was supported by Hungarian National Research Fund OTKA-K77748.

Some results of this paper have appeared in the Master thesis of the author (in Hungarian) [15].

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Szöllősi, F. Exotic complex Hadamard matrices and their equivalence. Cryptogr. Commun. 2, 187–198 (2010). https://doi.org/10.1007/s12095-010-0021-3

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