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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 |
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Vol. 17 (2023), No. 2, 359–396
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Abstract |
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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 -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 -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
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Mathematical Subject Classification
Primary: 05B10, 05E18
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Milestones
Received: 2 December 2020
Revised: 4 February 2022
Accepted: 4 April 2022
Published: 24 March 2023
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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 |
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| J. F. Dillon | |
| National Security Agency Fort George G Meade, MD United States |
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| J. Jedwab | |
| Department of Mathematics Simon Fraser University Burnaby, BC Canada |
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| T. Rabbani | |
| Department of Computer Science University of Maryland College Park, MD United States |
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| K. Smith | |
| Department of Mathematics and
Statistics Sam Houston State University Huntsville, TX United States |
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| W. Yolland | |
| Department of Mathematics Simon Fraser University Burnaby, BC Canada |
MetaOptima Technology Inc. Vancouver, BC Canada |
| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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