Abstract
We show by explicit construction that the equivalence classes of multiplicative central (p n, p n, p n, 1)-RDSs relative to \({\mathbb Z}_p^n\) in groups E with \(E/{\mathbb Z}_p^n \cong {\mathbb Z}_p^n\) are in one-to-one correspondence with the strong isotopism classes of presemifields of order p n. We also show there are 1,446 equivalence classes of central (16, 16, 16, 1)-RDS relative to \({\mathbb Z}_2^4\), in groups E for which \(E/{\mathbb Z}_2^4 \cong {\mathbb Z}_2^4\). Only one is abelian.
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Acknowledgements
This work forms part of the PhD thesis [10] of the first author, taken under the supervision of the second author. We would very much like to thank John Dillon for discovering and correcting errors in results in [19, Chapter 7], repeated in [13, §9.3.1.1] and [20], which we had quoted in an earlier version of this paper. We also thank the referees for excellent suggestions which greatly improved the clarity of our exposition.
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Farmer, D.G., Horadam, K.J. Equivalence classes of multiplicative central (p n, p n, p n, 1)-relative difference sets. Cryptogr. Commun. 3, 17–28 (2011). https://doi.org/10.1007/s12095-010-0026-y
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DOI: https://doi.org/10.1007/s12095-010-0026-y