Cocyclic Development of Designs
K.J. Horadam
and W. de Launey
DOI: 10.1023/A:1022403732401
Abstract
We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G \times G. Negacyclic and 
-cyclic development are both special cases of cocyclic development.
-cyclic development are both special cases of cocyclic development. Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic.
Pages: 267–290
Keywords: orthogonal design; Hadamard matrix; difference set; group development; negacyclic development; $ohgr$-cyclic development; cocycle; cohomology group; homology group; extension group
Full Text: PDF
References
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2. K.S. Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer-Verlag, New York, 1982.
3. W. de Launey, "On the construction of n-dimensional designs from 2-dimensional designs," Australas. J. Combin. 1 (1990), 67-81.
4. W. de Launey and K.J. Horadam, "A weak difference set construction for higher dimensional designs", Designs, Codes and Cryptography 3 (1993), 75-87.
5. P. Delsarte, J.M. Goethals, and J.J. Seidel, "Orthogonal matrices with zero diagonal II", Can. J. Math. 23 (1971), 816-832.
6. A.V. Geramita and J. Seberry, Orthogonal Designs, Lecture Notes in Pure and Appl. Math. 45, Dekker, New York, 1979.
7. P.J. Hilton and U. Stammbach, A Course in Homological Algebra, Graduate Texts in Math. 4, Springer-Verlag, New York, 1971.
8. D.L. Johnson, Presentation of Groups, London Math. Soc. Lecture Note Ser. 22, Cambridge University Press, Cambridge, 1976.
9. C. Miller, "The second homology group of a group; relations among commutators", Proc. Amer. Math. Soc. 3 (1952), 588-595.
10. C.T.C. Wall, "Resolutions for extensions of groups," Math. Proc. Cambridge Philos. Soc. 57 (1961), 251-255.
11. W.D. Wallis, A.P. Street, and J.S. Wallis, Combinatorics: Room Squares, Sum-free Sets, Hadamard Matrices, Lecture Notes in Math. 292, Springer-Verlag, Berlin, 1972.
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