Abstract
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 \(v \le 46\).
Similar content being viewed by others
References
Arasu K.T.: Sequences and arrays with desirable correlation properties. In: Crnković D., Tonchev V. (eds.) Information security, coding theory and related combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, pp. 136–171. Amsterdam, (2011).
Balonin,N. A., Đoković D. Ž.: Negaperiodic Golay Pairs and Hadamard Matrices, arxiv:1508.00640, (2015).
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).
Bruzda W., Tadej W., Życzkowski K.: http://chaos.if.uj.edu.pl/~karol/hadamard/.
Butson A.T.: Generalized Hadamard matrices. Proc. Am. Math. Soc. 13, 894–898 (1962).
Craigen R.: Constructions for orthogonal matrices, PhD thesis, University of Waterloo, March (1991).
Craigen R.: Complex Golay sequences. J. Combin. Math. Combin. Comput. 15, 161–169 (1994).
Craigen R., Holzmann W., Kharaghani H.: Complex Golay sequences: structure and applications. Discret. Math. 252, 73–89 (2002).
Davis J. A., Jedwab J.: Peak-to-Mean Power Control in OFDM, Golay Complementary Sequences and Reed-Muller codes, Techical Report HPL-97-158, HP Laboratories Bristol (1997).
Đoković D.Ž.: Good matrices of order 33, 35 and 127 exist. J. Combin. Math. Combin. Comput. 14, 145–152 (1993).
Đoković D.Ž., Kotsireas I.S.: Periodic Golay pairs of length 72. In: Colbourn C. (ed.) Algebraic Design Theory and Hadamard Matrices, vol. 133, pp. 83–92. Springer, NewYork (2015).
Egan R.: On equivalence of negaperiodic Golay pairs. Des. Codes Cryptogr. 85(3), 523–532 (2017).
Egan R., Flannery D.L., Catháin P.Ó.: Classifying cocyclic Butson Hadamard matrices. In: Colbourn C. (ed.) Algebraic Design Theory and Hadamard Matrices, vol. 133, pp. 93–106. Springer, NewYork (2015).
Frank R.L.: Polyphase complementary codes. IEEE Trans. Inf. Theory IT 26, 641–647 (1980).
Golay M.J.E.: Multislit spectrometry. J. Opt. Soc. Am. 39, 437–444 (1949).
Ito N.: On Hadamard groups IV. J. Algebra 234, 651–663 (2000).
Kamali F., Kharaghani H.: Dihedral Golay sequences. Aust. J. Combin. 18, 139–145 (1998).
Ma S.L., Ng W.S.: On non-existence of perfect and nearly perfect sequences. Int. J. Inf. Coding Theory 1(1), 15–38 (2009).
Paterson K.G.: Generalized Reed-Muller codes and power control in OFDM modulation. IEEE Trans. Inf. Theory 46(1), 104–120 (2000).
Schmidt B.: Williamson matrices and a conjecture of Ito’s. Des. Codes Cryptogr. 17, 61–68 (1999).
Sivaswamy R.: Multiphase complementary codes. IEEE Trans. Inf. Theory IT 24, 564–572 (1978).
Tadej W., Życzkowski K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13(2), 133–177 (2006).
Turyn R.J.: Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings. J. Combin. Theory A 16, 313–333 (1974).
Werner R.F.: All teleportation and dense coding schemes. J. Phys. A 34, 7081–7094 (2001).
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Jedwab.
Rights and permissions
About this article
Cite this article
Egan, R. Phased unitary Golay pairs, Butson Hadamard matrices and a conjecture of Ito’s. Des. Codes Cryptogr. 87, 67–74 (2019). https://doi.org/10.1007/s10623-018-0485-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0485-2