an:00786137
Zbl 0833.05013
Flannery, D. L.
Transgression and the calculation of cocyclic matrices
EN
Australas. J. Comb. 11, 67-78 (1995).
00027504
1995
j
05B20 20J99 20J06
transgression homomorphism; binary matrix; cocyclic Hadamard conjecture; cocyclic matrix; cocyclic Hadamard matrices; universal coefficient theorem; cohomology; dihedral group
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 \textit{W. de Launey} and \textit{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. \textit{W. de Launey}, \textit{K. J. Horadam} and \textit{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.
N.Ito (Nagoya-Tenpaku)
Zbl 0785.05019