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Equivalence classes of multiplicative central \((p^{n}, p^{n}, p^{n}, 1)\)-relative difference sets. (English) Zbl 1234.05047
Summary: 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 \(E/\mathbb Z_{p}^n\cong \mathbb Z_p^n\) arein 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.
MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B25 Combinatorial aspects of finite geometries
Software:
Magma
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