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The cocyclic Hadamard matrices of order less than 40. (English) Zbl 1246.05033
Summary: 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 W. de Launey, D. L. Flannery and K. J. Horadam [Discrete Appl. Math. 102, No.1–2, 47–61 (2000; Zbl 0956.05026)] 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.

MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
Software:
GAP; Magma; nauty
Full Text:
References:
 [1] Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 [2] de Launey W., Flannery D.L., Horadam K.J.: Cocyclic Hadamard matrices and difference sets. Discrete Appl. Math. 102(1–2), 47–61 (2000) · Zbl 0956.05026 · doi:10.1016/S0166-218X(99)00230-9 [3] Flannery D.L.: Cocyclic Hadamard matrices and Hadamard groups are equivalent. J. Algebra 192(2), 749–779 (1997) · Zbl 0889.05032 · doi:10.1006/jabr.1996.6949 [4] Horadam K.J.: Hadamard Matrices and their Applications. Princeton University Press, Princeton, NJ (2007) · Zbl 1145.05014 [5] Horadam K.J., de Launey W.: Cocyclic development of designs. J. Algebraic Combin. 2(3), 267–290 (1993) · Zbl 0785.05019 · doi:10.1023/A:1022403732401 [6] Ito N.: On Hadamard groups. J. Algebra 168(3), 981–987 (1994) · Zbl 0906.05012 · doi:10.1006/jabr.1994.1266 [7] Ito N., Okamoto T.: On Hadamard groups of order 72. Algebra Colloq. 3(4), 307–324 (1996) · Zbl 0869.05017 [8] Kharaghani H., Tayfeh-Rezaie B.: A Hadamard matrix of order 428. J. Combin. Des. 13(6), 435–440 (2005) · Zbl 1076.05017 · doi:10.1002/jcd.20043 [9] McKay B.: Nauty User’s Guide, Version 2.2 (2007). http://cs.anu.edu.au/$$\sim$$bdm/nauty/nug.pdf . [10] 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 . · Zbl 1242.05038 [11] Orrick W.P.: Switching operations for Hadamard matrices. SIAM J. Discrete Math. 22(1), 31–50 (2008) · Zbl 1156.05012 · doi:10.1137/050641727 [12] 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/ . [13] Röder M.: The quasiregular projective planes of order 16. Glasnik Matematicki. 43(2), 231–242 (2008) · Zbl 1151.05007 · doi:10.3336/gm.43.2.01 [14] Röder M.: RDS, Version 1.1 (2008). http://www.gap-system.org/Packages/rds.html . [15] Spence E.: Classification of Hadamard matrices of order 24 and 28. Discrete Math. 140(1–3), 185–243 (1995) · Zbl 0827.05014 · doi:10.1016/0012-365X(93)E0169-5 [16] The GAP Group.: GAP–Groups, Algorithms, and Programming, Version 4.4.12 (2008). http://www.gap-system.org .
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