Glasnik Matematicki, Vol. 58, No. 2 (2023), 247-257. \( \)

BUSH-TYPE BUTSON HADAMARD MATRICES

Hadi Kharaghani, Thomas Pender, Caleb Van't Land and Vlad Zaitsev

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge AB T1K 3M4, Canada
e-mail:kharaghani@uleth.ca

Department of Mathematics, University of Simon Fraser, Burnaby BC V5A 1S6, Canada
e-mail:tsp7@sfu.ca

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge AB T1K 3M4, Canada
e-mail:caleb.vantland@uleth.ca

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge AB T1K 3M4, Canada
e-mail:vlad.zaitsev@uleth.ca

Abstract.   Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type Butson Hadamard matrices over the group \(G\) of \(n\)-th roots of unity is introduced that is also valid over any subgroup of \(G\). The case of Bush-type Butson Hadamard matrices of even order will be discussed.

2020 Mathematics Subject Classification.   05B20

Key words and phrases.   Hadamard matrix, unbiased Hadamard matrix, Bush-type Hadamard matrix

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.58.2.07

References:

  1. D. Best and H. Kharaghani, Unbiased complex Hadamard matrices and bases, Cryptogr. Commun. 2 (2010), 199–209.
    MathSciNet    CrossRef

  2. T. Beth, D. Jungnickel and H. Lenz, Design theory. Vols. I and II, Cambridge University Press, Cambridge, 1999.
    MathSciNet    CrossRef

  3. K. A. Bush, An inner orthogonality of Hadamard matrices, J. Austral. Math. Soc. 12 (1971), 242–248.
    MathSciNet    CrossRef

  4. K. A. Bush, Unbalanced Hadamard matrices and finite projective planes of even order, J. Combinatorial Theory Ser. A 11 (1971), 38–44.
    MathSciNet    CrossRef

  5. A. T. Butson, Generalized Hadamard matrices, Proc. Amer. Math. Soc. 13 (1962), 894–898.
    MathSciNet    CrossRef

  6. A. T. Butson, Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Canadian J. Math. 15 (1963), 42–48.
    MathSciNet    CrossRef

  7. C. J. Colbourn and J. H. Dinitz, eds., Handbook of combinatorial designs, Chapman & Hall/CRC, Boca Raton, 2007.
    MathSciNet    CrossRef

  8. D. A. Drake, Partial \(\lambda\)-geometries and generalized Hadamard matrices over groups, Canadian J. Math. 31 (1979), 617–627.
    MathSciNet    CrossRef

  9. T. Durt, B.-G. Englert, I. Bengtsson and K. Życzkowski, On mutually unbiased bases, International journal of quantum information 8 (2010), 535–640.
    CrossRef

  10. R. Guy, H. Hanani, N. Sauer and J. Schönheim, eds., Combinatorial structures and their applications, Gordon and Breach, New York, 1970.
    MathSciNet    Link

  11. W. H. Holzmann, H. Kharaghani and W. Orrick, On the real unbiased Hadamard matrices, in: Combinatorics and graphs, Amer. Math. Soc., Providence, 2010, 243–250.
    MathSciNet    CrossRef

  12. K. J. Horadam, Hadamard matrices and their applications, Princeton University Press, Princeton, 2007.
    MathSciNet    CrossRef

  13. Y. J. Ionin and H. Kharaghani, Doubly regular digraphs and symmetric designs, J. Combin. Theory Ser. A 101 (2003), 35–48.
    MathSciNet    CrossRef

  14. Z. Janko and H. Kharaghani, A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs, J. Combin. Theory Ser. A 98 (2002), 118–126.
    MathSciNet    CrossRef

  15. H. Kharaghani, New class of weighing matrices, Ars Combin. 19 (1985), 69–72.
    MathSciNet

  16. T. Y. Lam and K. H. Leung, On vanishing sums of roots of unity, J. Algebra 224 (2000) 91–109.
    MathSciNet    CrossRef

  17. J. Seberry, Orthogonal designs. Hadamard matrices, quadratic forms and algebras, Springer, Cham, 2017.
    MathSciNet    CrossRef

  18. S. S. Shrikhande, Generalized Hadamard matrices and orthogonal arrays of strength two, Canadian J. Math. 16 (1964), 736–740.
    MathSciNet    CrossRef

  19. E. Verheiden, Hadamard matrices and projective planes, J. Combin. Theory Ser. A 32 (1982), 126–131.
    MathSciNet    CrossRef

  20. W. D. Wallis, On a problem of K. A. Bush concerning Hadamard matrices, Bull. Austral. Math. Soc. 6 (1972), 321–326.
    MathSciNet    CrossRef

  21. A. Winterhof, On the non-existence of generalized Hadamard matrices, J. Statist. Plann. Inference 84 (2000), 337–342.
    MathSciNet    CrossRef
Glasnik Matematicki Home Page