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On cocyclic weighing matrices and the regular group actions of certain Paley matrices. (English) Zbl 0961.05068
By definition a weighing matrix $$W= W(x,t)$$ is an $$n\times n$$ matrix with entries in $$\{0,\mp 1\}$$ satisfying $$WW^T= W^TW= tI_n$$ where $$T$$ denotes the transposition and $$I_n$$ is the identity matrix of degree $$n$$. An automorphism of $$W$$ is a pair of signed permutations $$(P,Q)$$ such that $$PMQ^T= M$$, and the set of all automorphisms of $$W$$ forms a group which is called the group of automorphisms of $$W$$ and is denoted by $$\operatorname{Aut}(W)$$. Since Paley conference matrices are weighing matrices a study of the group of automorphisms of such matrices is set up in the paper. The study is also carried out for certain Paley Hadamard matrices. The authors determine all regular group actions obtained from these Paley matrices.

##### MSC:
 05E20 Group actions on designs, etc. (MSC2000) 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Magma
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##### References:
 [1] Assmus, E.F.; Salwach, C.J., The (16,6,2) designs, Internat. J. math. math. sci., 2, 261-281, (1979) · Zbl 0424.05009 [2] Baliga, A.; Horadam, K.J., Cocyclic matrices over Zt × Z22, Austral. J. combin., 11, 123-134, (1995) · Zbl 0838.05017 [3] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system I: the user language, J. symb. comput., 24, 235-269, (1997) · Zbl 0898.68039 [4] Carlitz, L., A theorem on permutations in a finite field, Proc. amer. math. soc., 11, 456-459, (1960) · Zbl 0095.03003 [5] J. Conway, R. Curits, S. Norton, R. Parker, R. Wilson, Atlas of Finite Groups III, Clarendon Press, Oxford, 1985. [6] de Launey, W.; Flannery, D.; Horadam, K.J., Cocyclic Hadamard matrices and difference sets, Discrete appl. math., 102, 47-61, (2000) · Zbl 0956.05026 [7] W. de Launey, M.J. Smith, Cocyclic orthogonal designs and the asymptotic existence of maximal size relative difference sets with forbidden subgroup size 2, J. Combin. Theory Ser. A, to appear. · Zbl 1001.05032 [8] J.D. Dixon, B. Mortimer, Permutation Groups, Springer, New York, 1991. [9] Flannery, D.L., Transgression and the calculation of cocyclic matrices, Austral. J. combin., 11, 67-78, (1995) · Zbl 0833.05013 [10] Flannery, D.L., Calculation of cocyclic matrices, J. pure appl. algebra, 112, 181-190, (1996) · Zbl 0867.20043 [11] Flannery, D.L., Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. algebra, 192, 749-779, (1997) · Zbl 0889.05032 [12] Gilman, R.E., On the Hadamard determinant theorem and orthogonal determinants (matrices?), Bull. amer. math. soc., 37, 30-31, (1931) · JFM 57.0124.01 [13] Goethals, J.M.; Seidel, J.J., Orthogonal matrices with zero diagonal, Canad. J. math., 19, 1001-1010, (1967) · Zbl 0155.35601 [14] Hall, M., Note on the Mathieu group M_12, Arch. math., 13, 334-340, (1962) · Zbl 0109.25704 [15] K.J. Horadam, W. de Launey, Cocyclic development of designs, J. Algebraic Combin. 2 (1993) 267-290, Erratum 3 (1994) 129. · Zbl 0785.05019 [16] B. Huppert, N. Blackburn, Finite Groups II, Springer, Berlin, 1982. · Zbl 0477.20001 [17] B. Huppert, N. Blackburn, Finite Groups III, Springer, Berlin, 1982. · Zbl 0514.20002 [18] Ito, N., Note on Hadamard groups of quadratic residue type, Hokkaido math. J., 22, 373-378, (1993) · Zbl 0793.05028 [19] Ito, N., On Hadamard groups, J. algebra, 168, 981-987, (1994) · Zbl 0906.05012 [20] Ito, N., On Hadamard groups II, J. algebra, 169, 936-942, (1994) · Zbl 0808.05016 [21] Kantor, W.M., Automorphism groups of Hadamard matrices, J. combin. theory ser. A, 6, 279-281, (1969) · Zbl 0206.02103 [22] R.C. Mullin, A note on balanced weighing matrices, Proceedings of Third Australian Conference on Combinatorial Mathematics, Brisbane, 1974. · Zbl 0308.05020 [23] Paley, R.E.A.C., On orthogonal matrices, J. math. phys., 12, 311-320, (1933) · Zbl 0007.10004 [24] 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 [25] D. Raghavarao, Constructions and combinatorial problems in the design of experiments, Wiley, New York, 1971. · Zbl 0222.62036 [26] S.T. Schibell, private communication, 1993. [27] Turyn, R.J., An infinite class of williamson matrices, J. combin. theory ser. A, 12, 319-321, (1972) · Zbl 0237.05008 [28] Yamada, M., Hadamard matrices of generalized quaternion type, Discrete math., 87, 187-196, (1991) · Zbl 0725.05028 [29] Yamamoto, K., On a generalized williamson equation, Colloquia math. soc. janos bolyai, 37, 839-850, (1985)
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