Hadamard matrices and dihedral groups
type: Item
instance of: scholarly article
title: Hadamard matrices and dihedral groups
zbMATH Open document ID: 0859.05031
DOI: 10.1023/A:1027342024177
author: Hiroshi Kimura
published in: Designs, Codes and Cryptography
publication date: 1997-04-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 934919
zbMATH Keywords: Hadamard matrix
zbMATH Keywords: dihedral group
reviewed by: D. Raghavarao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 934919