Hermitian congruence and the existence of completion of generalized Hadamard matrices
type: Item
instance of: scholarly article
title: Hermitian congruence and the existence of completion of generalized Hadamard matrices
zbMATH Open document ID: 0667.05011
DOI: 10.1016/0097-3165(88)90054-4
author: Bradley W. Brock
published in: Journal of Combinatorial Theory. Series A
publication date: 1988-01-01T00:00:00Z
review text: Some problems on existence, construction and application of generalized Hadamard matrices (GHM) in the group ring image \[ R=Q(G)/(\sum_{g\in G}g), \] where G is a finite group, are considered. The matrix equation \[ AA^*=A^*A=mI_ h+\mu J_ h \] in the cyclotomic field and in the group ring R is investigated. It is shown that for \(m=h\) and \(\mu =0\) the obtained results yield a nonexistence theorem for GHMs. Necessary and sufficient conditions for the existence of those matrices are given. The results of Marshall Hall an Hadamard completions are extended to GHMs. The application of GHM to coding theory (construction of codes attaining the Plotkin bound) is also considered. A theorem on the automorphism group of a GHM is given.
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 4091519
zbMATH Keywords: Hermitian congruence
zbMATH Keywords: generalized Hadamard matrices
zbMATH Keywords: GHM
reviewed by: S. S. Agayan
MaRDI profile type: Publication
full work available at URL: 0097-3165(88)90054-4
OpenAlex ID: W2012968160
cites work: Diophantine Equations in Cyclotomic Fields
cites work: On collineation groups of symmetric block designs
cites work: Orthogonal Arrays of Strength two and three
cites work: Generalized Hadamard Matrices
cites work: Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences
cites work: Q3892261
cites work: A construction for generalized Hadamard matrices GH(4q,EA(q))
cites work: Q3309841
cites work: On the non-existence of generalized Hadamard matrices
cites work: Tri-weight codes and generalized hadamard matrices
cites work: Partial λ-Geometries and Generalized Hadamard Matrices Over Groups
cites work: Projective Planes
cites work: Q4152661
cites work: Normal Completions of Incidence Matrices
cites work: Orthomorphisms of Groups and Orthogonal Latin Squares. I
cites work: Q5801410
cites work: On difference matrices, resolvable transversal designs and generalized Hadamard matrices
cites work: On difference matrices and regular latin squares
cites work: Q4146667
cites work: Number fields
cites work: Generalized Hadamard matrices and related matrices and their applications to the construction of the positive definite integral Hermitian forms
cites work: Q3851094
cites work: Q3872477
cites work: A construction for generalized Hadamard matrices
cites work: On the Problem of Construction of Orthogonal Arrays
cites work: Generalized Hadamard Matrices and Orthogonal Arrays of Strength Two
cites work: Q3895487
cites work: Q5556895
cites work: Integral and rational completions of combinatorial matrices
cites work: Complex Hadamard matrices
cites work: Combinatorics. Room squares, sum-free sets, Hadamard matrices
description: scientific article; zbMATH DE number 4091519