Difference sets and doubly transitive actions on Hadamard matrices

Cathain, Padraig O. (2012) Difference sets and doubly transitive actions on Hadamard matrices. Journal of Combinatorial Theory Series A, 119 6: 1235-1249. doi:10.1016/j.jcta.2012.02.011

Attached Files (Some files may be inaccessible until you login with your UQ eSpace credentials)
Name Description MIMEType Size Downloads

Author Cathain, Padraig O.
Title Difference sets and doubly transitive actions on Hadamard matrices
Journal name Journal of Combinatorial Theory Series A   Check publisher's open access policy
ISSN 0097-3165
1096-0899
Publication date 2012-08-01
Sub-type Article (original research)
DOI 10.1016/j.jcta.2012.02.011
Volume 119
Issue 6
Start page 1235
End page 1249
Total pages 15
Place of publication Maryland Heights United States
Publisher Academic Press
Language eng
Formatted abstract
Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon.

We then use our classification to show that the only cocyclic Hadamard matrices developed from a difference set with non-affine automorphism group are those that arise from the Paley Hadamard matrices.

If H is a cocyclic Hadamard matrix developed from a difference set then the automorphism group of H is doubly transitive. We classify all difference sets which give rise to Hadamard matrices with non-affine doubly transitive automorphism group. A key component of this is a complete list of difference sets corresponding to the Paley Hadamard matrices. As part of our classification we uncover a new triply infinite family of skew-Hadamard difference sets. To our knowledge, these are the first skew-Hadamard difference sets to be discovered in non-abelian p-groups with no exponent restriction.

As one more application of our main classification, we show that Hallʼs sextic residue difference sets give rise to precisely one cocyclic Hadamard matrix.
Keyword Hadamard matrix
Paley-Hadamard design
Skew-Hadamard difference set
Doubly transitive permutation group
Automorphism-Groups
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
Versions
Version Filter Type
Citation counts: TR Web of Science Citation Count  Cited 1 times in Thomson Reuters Web of Science Article | Citations
Scopus Citation Count Cited 1 times in Scopus Article | Citations
Google Scholar Search Google Scholar
Created: Sat, 08 Jun 2013, 06:07:08 EST by System User on behalf of Mathematics