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

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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 \(v \le 46\).

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References

  1. 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).

  2. Balonin,N. A., Đoković D. Ž.: Negaperiodic Golay Pairs and Hadamard Matrices, arxiv:1508.00640, (2015).

  3. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruzda W., Tadej W., Życzkowski K.: http://chaos.if.uj.edu.pl/~karol/hadamard/.

  5. Butson A.T.: Generalized Hadamard matrices. Proc. Am. Math. Soc. 13, 894–898 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  6. Craigen R.: Constructions for orthogonal matrices, PhD thesis, University of Waterloo, March (1991).

  7. Craigen R.: Complex Golay sequences. J. Combin. Math. Combin. Comput. 15, 161–169 (1994).

    MathSciNet  MATH  Google Scholar 

  8. Craigen R., Holzmann W., Kharaghani H.: Complex Golay sequences: structure and applications. Discret. Math. 252, 73–89 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  9. 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).

  10. Đoković D.Ž.: Good matrices of order 33, 35 and 127 exist. J. Combin. Math. Combin. Comput. 14, 145–152 (1993).

  11. 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).

  12. Egan R.: On equivalence of negaperiodic Golay pairs. Des. Codes Cryptogr. 85(3), 523–532 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  13. 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).

    Chapter  Google Scholar 

  14. Frank R.L.: Polyphase complementary codes. IEEE Trans. Inf. Theory IT 26, 641–647 (1980).

    Article  MathSciNet  Google Scholar 

  15. Golay M.J.E.: Multislit spectrometry. J. Opt. Soc. Am. 39, 437–444 (1949).

    Article  Google Scholar 

  16. Ito N.: On Hadamard groups IV. J. Algebra 234, 651–663 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  17. Kamali F., Kharaghani H.: Dihedral Golay sequences. Aust. J. Combin. 18, 139–145 (1998).

    MathSciNet  MATH  Google Scholar 

  18. 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).

    Article  MathSciNet  MATH  Google Scholar 

  19. Paterson K.G.: Generalized Reed-Muller codes and power control in OFDM modulation. IEEE Trans. Inf. Theory 46(1), 104–120 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  20. Schmidt B.: Williamson matrices and a conjecture of Ito’s. Des. Codes Cryptogr. 17, 61–68 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  21. Sivaswamy R.: Multiphase complementary codes. IEEE Trans. Inf. Theory IT 24, 564–572 (1978).

    Article  MATH  Google Scholar 

  22. Tadej W., Życzkowski K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13(2), 133–177 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  23. 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).

    Article  MathSciNet  MATH  Google Scholar 

  24. Werner R.F.: All teleportation and dense coding schemes. J. Phys. A 34, 7081–7094 (2001).

    Article  MathSciNet  MATH  Google Scholar 

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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.

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  1. Department of Mathematics, University of Rijeka, Rijeka, Croatia

    Ronan Egan

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  1. Ronan Egan
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Correspondence to Ronan Egan.

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Communicated by J. Jedwab.

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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

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  • DOI: https://doi.org/10.1007/s10623-018-0485-2

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