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Constructions of difference sets in nonabelian 2-groups

T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith and W. Yolland

Vol. 17 (2023), No. 2, 359–396
Abstract

Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of 2-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining 2-groups. In particular, we determine that all groups of order 256 not excluded by the two classical nonexistence criteria contain a difference set, in agreement with previous findings for groups of order 4, 16, and 64. We provide suggestions for how the existence question for difference sets in 2-groups of all orders might be resolved.

Keywords
difference set, nonabelian, 2-group, construction
Mathematical Subject Classification
Primary: 05B10, 05E18
Milestones
Received: 2 December 2020
Revised: 4 February 2022
Accepted: 4 April 2022
Published: 24 March 2023
Authors
T. Applebaum
Department of Mathematics and Statistics
University of Richmond
Richmond, VA
United States
DeepMind
London
United Kingdom
J. Clikeman
Department of Mathematics and Statistics
University of Richmond
Richmond, VA
United States
Google
Mountain View, CA
United States
J. A. Davis
Department of Mathematics and Statistics
University of Richmond
Richmond, VA
United States
J. F. Dillon
National Security Agency
Fort George G Meade, MD
United States
J. Jedwab
Department of Mathematics
Simon Fraser University
Burnaby, BC
Canada
T. Rabbani
Department of Computer Science
University of Maryland
College Park, MD
United States
K. Smith
Department of Mathematics and Statistics
Sam Houston State University
Huntsville, TX
United States
W. Yolland
Department of Mathematics
Simon Fraser University
Burnaby, BC
Canada
MetaOptima Technology Inc.
Vancouver, BC
Canada

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