Abstract
Reversible difference sets have been studied extensively by many people. Dillon showed that reversible difference sets existed in groups \({(C_{2^{r}})^{2}}\) and C4. Davis and Polhill showed the existence of DRAD difference sets in the groups \({(C_{2^{r}})^{2}}\) for \({r\geq 2}\) and also for the group C4. This paper gives a construction technique utilizing character values, rational idempotents, and tiles to produce both reversible and DRAD Hadamard difference sets in the group \({C_{2^{r}} \times C_{2^{r}}}\) for \({r\geq 2}\) and in C4. We also show necessary conditions for both reversible and DRAD difference sets in abelian 2-groups.
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Acknowledgments
This work was done as partial fulfillment of a Ph.D. at Central Michigan University under the guidance of Ken W. Smith. The author is extremely grateful for the help and support of both Ken W. Smith and Central Michigan University.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Webster, J.D. Reversible difference sets with rational idempotents. Arab. J. Math. 2, 103–114 (2013). https://doi.org/10.1007/s40065-012-0055-9
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DOI: https://doi.org/10.1007/s40065-012-0055-9