Williamson Hadamard matrices and Gauss sums
type: Item
instance of: scholarly article
title: Williamson Hadamard matrices and Gauss sums
zbMATH Open document ID: 0584.05016
author: Koichi Yamamoto
author: Mieko Yamada
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: In this paper, the study of the well-known Williamson equation is continued for constructing Hadamard matrices of Williamson type. Some important special cases of the equation are investigated. The main result is contained in the following theorem: Let n be odd and \(q=2n-1\) be a power of prime. Let K be a quadratic extension of the finite field \(F=GF(g)\), \(\psi\) be the quadratic residue character of F and \(\alpha\) be an element of K such that \(2F^*\) generates the quotient group \(K^*/F^*\). Let's define \(z_ m\) \((m=1,2,...,n-1):\) \[ z_ m=(\psi (2S_{K/F}\alpha^{4m})-i\psi (2S_{K/F}\alpha^{4m-n}))/(1-i), \] where \(S_{K/F}\) is the relative trace in K/F. Then \(H=wI+\sum^{n- 1}_{m=1}z_ mT^ m\) is a symmetric circulant quaternion Hadamard matrix of order n where I is the unit matrix and T is the basic circulant matrix \((T^ n=I)\), \(w=(1+i+j+k)/2\), 1,i,j,k are the quaternion units.
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 3935055
zbMATH Keywords: Gauss sums
zbMATH Keywords: Williamson equation
zbMATH Keywords: Hadamard matrices
reviewed by: S. S. Agayan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03740703
OpenAlex ID: W2024706349
DOI: 10.2969/JMSJ/03740703
Recommended article: Q4459073
Recommended article: Q4534480
Recommended article: Q5295613
Recommended article: Hadamard matrices of Williamson type: A challenge for computer algebra
Recommended article: Quadratic Gauss sums on matrices
Recommended article: Gauss sums and quadratic forms on matrices
Recommended article: Gauss sums over some matrix groups
Recommended article: Williamson type Hadamard matrices with circulant components
Recommended article: Q4545592
Recommended article: On Hadamard matrices
description: scientific article; zbMATH DE number 3935055