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Cocyclic Hadamard matrices and difference sets. (English) Zbl 0956.05026
The authors explain the connection between (relative) difference sets, cocyclic Hadamard matrices, cocycles, coboundaries and perfect binary arrays. In particular, they mention the interesting conjecture that for any $$t$$, cocyclic Hadamard matrices of order $$4t$$ exist. The authors show that many of the known families of Hadamard matrices are actually families of cocyclic Hadamard matrices.

MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 20J06 Cohomology of groups 05B05 Combinatorial aspects of block designs
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References:
 [1] E.F. Assmus, J.D. Key, Designs and their Codes, CUP, Cambridge, 1992. · Zbl 0762.05001 [2] Baliga, A.; Horadam, K.J., Cocyclic matrices over $$Zt × Z2\^{}\{2\}$$, Australas. J. combin., 11, 123-134, (1995) · Zbl 0838.05017 [3] T. Beth, D. Jungnickel, H. Lenz, Design Theory, CUP, Cambridge, 1993. [4] Chan, Y.K.; Siu, M.K.; Tong, P., Two-dimensional binary arrays with good autocorrelation, Inform. and control, 42, 125-130, (1979) · Zbl 0421.05010 [5] C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996. · Zbl 0836.00010 [6] de Launey, W., On the construction of $$n$$-dimensional designs from $$2$$-dimensional designs, Australas. J. combin., 1, 67-81, (1990) · Zbl 0758.05034 [7] W. de Launey, Cocyclic Hadamard matrices and relative difference sets, Hadamard Centenary Conference, U. Wollongong, Australia, December 1993, unpublished. [8] W. de Launey, On a family of cocyclic Hadamard matrices, preprint. · Zbl 1029.05025 [9] de Launey, W.; Horadam, K.J., A weak difference set construction for higher dimensional designs, Des. codes cryptogr., 3, 75-87, (1993) · Zbl 0838.05019 [10] W. de Launey, M.J. Smith, Cocyclic orthogonal designs and the asymptotic existence of maximal size relative difference sets with forbidden subgroup of size 2, 1998, preprint. · Zbl 1001.05032 [11] de Launey, W.; Stafford, R.M., On cocyclic weighing matrices and the regular actions of certain Paley matrices, Discrete appl. math., 102, 63-101, (2000) · Zbl 0961.05068 [12] Elliott, J.E.H.; Butson, A.T., Relative difference sets, Illinois J. math., 10, 517-531, (1966) · Zbl 0145.01503 [13] Flannery, D.L., Transgression and the calculation of cocyclic matrices, Australas. J. combin., 11, 67-78, (1995) · Zbl 0833.05013 [14] Flannery, D.L., Calculation of cocyclic matrices, J. pure appl. algebra, 112, 181-190, (1996) · Zbl 0867.20043 [15] Flannery, D.L., Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. algebra, 192, 749-779, (1997) · Zbl 0889.05032 [16] Goethals, J.-M.; Seidel, J.J., Orthogonal matrices with zero diagonal, Canad. J. math., 19, 1001-1010, (1967) · Zbl 0155.35601 [17] Goethals, J.-M.; Seidel, J.J., A skew Hadamard matrix of order $$36$$, J. austral. math. soc., 11, 343-344, (1970) · Zbl 0226.05015 [18] Holzmann, W.H.; Kharaghani, H., A computer search for complex golay sequences, Australas. J. combin., 10, 251-258, (1994) · Zbl 0818.05022 [19] Horadam, K.J., Progress in cocyclic matrices, Congr. numer., 118, 161-171, (1996) · Zbl 0898.05011 [20] K.J. Horadam, W. de Launey, Cocyclic development of designs, J. Algebraic Combin. 2 (1993) 267-290, Erratum 3 (1994) 129. · Zbl 0785.05019 [21] K.J. Horadam, W. de Launey, Generation of cocyclic Hadamard matrices, in: W. Bosma, A. van der Poorten (Eds.), Computational Algebra and Number Theory, Kluwer Academic, Dordrecht, 1995. · Zbl 0838.05018 [22] Horadam, K.J.; Lin, C., Construction of proper higher dimensional Hadamard matrices from perfect binary arrays, J. combin. math. combin. comput., 28, 237-248, (1998) · Zbl 0917.05016 [23] Ito, N., Note on Hadamard matrices of type $$Q$$, Stud. sci. math. hungar., 16, 389-393, (1981) · Zbl 0537.05012 [24] Ito, N., Note on Hadamard groups of quadratic residue type, Hokkaido math. J., 22, 373-378, (1993) · Zbl 0793.05028 [25] Ito, N., On Hadamard groups, J. algebra, 168, 981-987, (1994) · Zbl 0906.05012 [26] Ito, N., On Hadamard groups II, J. algebra, 169, 936-942, (1994) · Zbl 0808.05016 [27] N. Ito, Some remarks on Hadamard groups, in: Groups-Korea ’94, Walter de Gruyter, Berlin, 1995. [28] Ito, N., Remarks on Hadamard groups, Kyushu J. math., 50, 83-91, (1996) · Zbl 0889.05033 [29] Ito, N., On Hadamard groups III, Kyushu J. math., 51, 1-11, (1997) [30] Jedwab, J., Generalised perfect arrays and menon difference sets, Des. codes cryptogr., 2, 19-68, (1992) · Zbl 0767.05030 [31] Jungnickel, D., On automorphism groups of divisible designs, Canad. J. math., 24, 257-297, (1982) · Zbl 0465.05011 [32] Kopilovich, L.E., On perfect binary arrays, Electron. lett., 24, 566-567, (1988) · Zbl 0686.05015 [33] Perera, A.A.I.; Horadam, K.J., Cocyclic generalised Hadamard matrices and central relative difference sets, Des. codes cryptogr., 15, 187-200, (1998) · Zbl 0919.05007 [34] A. Pott, Finite Geometry and Character Theory, Lecture Notes in Mathematics, vol. 1601, Springer, Berlin, 1995. · Zbl 0818.05001 [35] B. Schmidt, Williamson matrices and a conjecture of Ito’s, preprint, September 1998. [36] Turyn, R.J., An infinite class of williamson matrices, J. combin. theory ser. A, 12, 319-321, (1972) · Zbl 0237.05008 [37] Williamson, J., Hadamard’s determinant theorem and the sum of four squares, Duke math J., 11, 65-81, (1944) · Zbl 0060.03202 [38] Yamada, M., Hadamard matrices of generalised quaternion type, Discrete math., 87, 187-196, (1991) · Zbl 0725.05028
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