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

摘要

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 k2. 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 v1, 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 v46.