Note on GMW designs
type: Item
instance of: scholarly article
title: Note on GMW designs
zbMATH Open document ID: 0964.05015
published in: European Journal of Combinatorics
publication date: 2001-07-08T00:00:00Z
review text: The classical difference sets with the classical parameters \[ ((q^n-1)/(q-1), (q^{n-1}-1)/(q-1), (q^{n-2}-1)/(q-1)) \] are the Singer difference sets (\(q\) is a prime power). The corresponding symmetric design is the classical point-hyperplane design of the \(n-1\)-dimensional projective space. For arbitrary prime powers \(q\), only one more general construction of difference sets with the classical parameters is known (GMW construction, see \textit{B. Gordon, W. H. Mills} and \textit{L. R. Welch} [Some new difference sets, Can. J. Math. 14, 614-625 (1962; Zbl 0111.24201)]). It is an old question to decide whether inequivalent GMW difference sets yield nonisomorphic designs. In this short but highly nontrivial paper, Kantor shows that isomorphism implies equivalence. The proof uses group theory, and therefore seems to be inappropriate for proving this (geometric) result (in the words of the author, ``using a cannon to kill an ant'').
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 51E05
zbMATH DE Number: 1567134
zbMATH Keywords: difference set
zbMATH Keywords: nonisomorphic designs
author: William M. Kantor
reviewed by: Alexander Pott
MaRDI profile type: Publication
OpenAlex ID: W2044905112
cites work: Some New Difference Sets
cites work: Q5577154
cites work: A characterization of Hadamard designs with SL\((2,q)\) acting transitively
cites work: On GMW designs and cyclic Hadamard designs
cites work: Homogeneous designs and geometric lattices
cites work: Finite geometry and character theory
cites work: Flag-transitive subgroups of Chevalley groups
cites work: Generators, relations and coverings of algebraic groups. II
cites work: Two-graphs and doubly transitive groups
cites work: Q5512231
DOI: 10.1006/EUJC.2000.0426
description: scientific article; zbMATH DE number 1567134