Abstract
In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997)
de Launey W., Flannery D.L., Horadam K.J.: Cocyclic Hadamard matrices and difference sets. Discrete Appl. Math. 102(1–2), 47–61 (2000)
Flannery D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192(2), 749–779 (1997)
Horadam K.J.: Hadamard Matrices and their Applications. Princeton University Press, Princeton, NJ (2007)
Horadam K.J., de Launey W.: Cocyclic development of designs. J. Algebraic Combin. 2(3), 267–290 (1993)
Ito N.: On Hadamard groups. J. Algebra 168(3), 981–987 (1994)
Ito N., Okamoto T.: On Hadamard groups of order 72. Algebra Colloq. 3(4), 307–324 (1996)
Kharaghani H., Tayfeh-Rezaie B.: A Hadamard matrix of order 428. J. Combin. Des. 13(6), 435–440 (2005)
McKay B.: Nauty User’s Guide, Version 2.2 (2007). http://cs.anu.edu.au/~bdm/nauty/nug.pdf.
Ó Catháin P.: Group Actions on Hadamard matrices. M.Litt. Thesis, National University of Ireland, Galway (2008). http://www.maths.nuigalway.ie/padraig/research.shtml.
Orrick W.P.: Switching operations for Hadamard matrices. SIAM J. Discrete Math. 22(1), 31–50 (2008)
Röder M.: Quasiregular Projective Planes of Order 16—A Computational Approach. PhD thesis, Technische Universität Kaiserslautern (2006). http://kluedo.ub.uni-kl.de/volltexte/2006/2036/.
Röder M.: The quasiregular projective planes of order 16. Glasnik Matematicki. 43(2), 231–242 (2008)
Röder M.: RDS, Version 1.1 (2008). http://www.gap-system.org/Packages/rds.html.
Spence E.: Classification of Hadamard matrices of order 24 and 28. Discrete Math. 140(1–3), 185–243 (1995)
The GAP Group.: GAP—Groups, Algorithms, and Programming, Version 4.4.12 (2008). http://www.gap-system.org.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.D. Key.
Rights and permissions
About this article
Cite this article
Ó Catháin, P., Röder, M. The cocyclic Hadamard matrices of order less than 40. Des. Codes Cryptogr. 58, 73–88 (2011). https://doi.org/10.1007/s10623-010-9385-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-010-9385-9