Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 67–74 | Cite as

Phased unitary Golay pairs, Butson Hadamard matrices and a conjecture of Ito’s

  • Ronan EganEmail author


Pairs of complementary binary or quaternary sequences of length v such as Golay pairs, complex Golay pairs and periodic Golay pairs may be used to construct Hadamard matrices and complex Hadamard matrices of order 2v. We generalize these and define unitary Golay pairs and phased unitary Golay pairs of length v with entries in the kth roots of unity for any \(k \ge 2\). This leads to a construction of Butson Hadamard matrices of order 2v over the kth roots of unity for even k. Ito conjectured that a central relative (4v, 2, 4v, 2v)-difference set exists in a dicyclic group of order 8v for all \(v \ge 1\), and this is known to imply the Hadamard conjecture. With our construction we prove that Ito’s conjecture also implies the stronger complex Hadamard conjecture. As a consequence, with this method we construct a complex Hadamard matrix of order 2v for any v for which Ito’s conjecture is verified, in particular, any \(v \le 46\).


Unitary Golay pair Butson Hadamard matrix Complementary sequences 

Mathematics Subject Classification

05B20 11B99 15B33 



This work has been fully supported by Croatian Science Foundation under the project 1637. The author would like to thank the reviewers for helpful suggestions that improved the quality of this paper.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RijekaRijekaCroatia

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