Hadamard matrices and dihedral groups
An \(n\times n\) matrix with elements \(0\) and \(1\) is called an Hadamard matrix, if the matrix \(H\) obtained from it by changing \(0\)'s to \(-1\)'s satisfies \(HH'=nI_n\). Let \(\text{D}_{2p}\) be a dihedral group of order \(2p\), where \(p\) is an odd integer and \(\mathbb{Z}\text{D}_{2p}\) be the group ring of \(\text{D}_{2p}\) over the ring \(\mathbb{Z}\) of integers. Four matrices \(A\), \(B\), \(C\), \(D\) each of order \(2p\times 2p\) over \(\mathbb{Z}\text{D}_{2p}\) are found such that \[ \begin{matrix} 1 & 1 & 1 & 1 & 1_{2p}' & 1_{2p}' & 1_{2p}' & 1_{2p}'\\ 1 & 1 & 0 & 0 & 1_{2p}' & 1_{2p}' & 0' & 0'\\ 1 & 0 & 1 & 0 & 1_{2p}' & 0' & 1_{2p}' & 0'\\ 1 & 0 & 0 & 1 & 0' & 1_{2p}' & 1_{2p}' & 0'\\ 1_{2p} & 1_{2p} & 1_{2p} & 1_{2p} & A & B & C & D\\ 1_{2p} & 1_{2p} & 0 & 1_{2p} & \overline B & A & D & \overline C\\ 1_{2p} & 0 & 1_{2p} & 1_{2p} & \overline C & \overline D & A & B\\ 1_{2p} & 0 & 0 & 0 & D & \overline C & B & \overline A\end{matrix} \] is a Hadamard matrix for \(p= 7, 9, 13, 17\), and 41, where \(1_{2p}\) is a column vector of \(2p\) ones, \(\overline X=J-X\) for \(X= A,B,C,D\) and for \(J\) a \(2p\times 2p\) matrix of ones. A general solution to the problem is still open.
scientific article; zbMATH DE number 934919