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Transgression and the calculation of cocyclic matrices. (English) Zbl 0833.05013
Let $$G$$ be a group of order $$4t$$. A function $$f$$ from $$G\times G$$ to $$Z_2$$ is cocyclic, if $$f$$ satisfies the following equations: $$f(a, b)f(ab, c)= f(b, c) f(a, bc)$$ for all elements $$a$$, $$b$$ and $$c$$ of $$G$$. A matrix whose entries are $$\pm 1$$ is called binary. A binary matrix $$M$$ labelled by the elements of $$G$$ is called cocylic (over $$G$$), if there exist a function $$g$$ from $$G$$ to $$Z_2$$ and a cocyclic function $$f$$ from $$G\times G$$ to $$Z_2$$ such that $$M= (f(a, b) g(ab))$$. The following cocyclic Hadamard conjecture was made by W. de Launey and K. J. Horadam in [Des. Codes Cryptography 3, No. 1, 75-87 (1993)]: For all $$t\geq 1$$, there is a cocyclic matrix of degree $$4t$$ that is Hadamard. W. de Launey, K. J. Horadam and A. Baliga presented methods to calculate cocyclic Hadamard matrices over abelian groups; see Des. Codes Cryptography 3, No. 1, 75-87 (1993), J. Algebr. Comb. 2, No. 3, 267-290 (1993; Zbl 0785.05019), and Australas. J. Comb. 11, 123-134 (1995).
In the present paper, the author presents a method to calculate so-called representative cocyclic Hadamard matrices over not necessarily abelian groups utilizing more standard results (including the universal coefficient theorem) in the cohomology theory of finite groups. In particular, the author gives the explicit calculation in the case of a dihedral group of order 8.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 20J99 Connections of group theory with homological algebra and category theory 20J06 Cohomology of groups