Abstract
We classify all the cocyclic Butson Hadamard matrices BH(n, p) of order n over the pth roots of unity for an odd prime p and n p ≤ 100. That is, we compile a list of matrices such that any cocyclic BH(n, p) for these n, p is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.
This paper is in final form and no similar paper has been or is being submitted elsewhere.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997)
Brock, B.: A new construction of circulant GH(p 2; Z p ). Discrete Math. 112(1–3), 249–252 (1993)
Bruzda, W., Tadej, W., Życzkowski, K.: http://chaos.if.uj.edu.pl/~karol/hadamard/
Butson, A.T.: Generalized Hadamard matrices. Proc. Amer. Math. Soc. Soc. 13, 894–898 (1962)
Cooke, C.H., Heng, I.: On the non-existence of some generalised Hadamard matrices. Australas. J. Combin. 19, 137–148 (1999)
de Launey, W.: On the nonexistence of generalised weighing matrices. Ars Comb. 17, 117–132 (1984)
de Launey, W.: Circulant GH(p 2; Z p ) exist for all primes p. Graphs Combin. 8(4), 317–321 (1992)
de Launey, W.: Generalised Hadamard matrices which are developed modulo a group. Discrete Math. 104, 49–65 (1992)
de Launey, W., Flannery, D.L.: Algebraic Design Theory. Mathematical Surveys and Monographs, vol. 175. American Mathematical Society, Providence, RI (2011)
Egan, R., Flannery, D.L., Ó Catháin, P.: http://www.maths.nuigalway.ie/~dane/BHIndex.html
Flannery, D. L., Egan, R.: On linear shift representations. J. Pure Appl. Algebra 219(8), 3482–3494 (2015)
Flannery, D.L., O’Brien, E.A.: Computing 2-cocycles for central extensions and relative difference sets. Comm. Algebra 28, 1939–1955 (2000)
The GAP Group: GAP – Groups, Algorithms, and Programming. http://www.gap-system.org
Hiranandani, G., Schlenker, J.-M.: Small circulant complex Hadamard matrices of Butson type. http://arxiv.org/abs/1311.5390
Hirasaka, M., Kim, K.-T., Mizoguchi, Y.: Uniqueness of Butson Hadamard matrices of small degrees. http://arxiv.org/abs/1402.6807
Horadam, K.J.: Hadamard Matrices and Their Applications. Princeton University Press, Princeton (2007)
Karpilovsky, G.: The Schur Multiplier. London Mathematical Society Monographs. New Series, 2. The Clarendon Press/Oxford University Press, New York (1987)
Lam, T.Y., Leung, K.H.: On vanishing sums of roots of unity. J. Algebra 224(1), 91–109 (2000)
McKay, B., Piperno, A.: http://pallini.di.uniroma1.it/
McNulty, D., Weigert, S.: Isolated Hadamard matrices from mutually unbiased product bases. J. Math. Phys. 53(12), 122202, 16 pp (2012)
Ó Catháin, P., Röder, M.: The cocyclic Hadamard matrices of order less than 40. Des. Codes Cryptogr. 58(1), 73–88 (2011)
Perera, A.A.I., Horadam, K.J.: Cocyclic generalised Hadamard matrices and central relative difference sets. Des. Codes Cryptogr. 15, 187–200 (1998)
Röder, M.: The GAP package RDS. http://www.gap-system.org/Packages/rds.html
Röder, M.: Quasiregular projective planes of order 16—a computational approach. Ph.D. Thesis, Technische Universität Kaiserslautern (2006)
Winterhof, A.: On the nonexistence of generalized Hadamard matrices. J. Statist. Plann. Inference 84, 337–342 (2000)
Acknowledgements
R. Egan received funding from the Irish Research Council (Government of Ireland Postgraduate Scholarship). P. Ó Catháin was supported by Australian Research Council grant DP120103067.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Hadi Kharaghani on the occasion of his 70th birthday
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Egan, R., Flannery, D., Catháin, P.Ó. (2015). Classifying Cocyclic Butson Hadamard Matrices. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-17729-8_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17728-1
Online ISBN: 978-3-319-17729-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)