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A new construction of central relative \((p^a,p^a,p^a,1)\)-difference sets. (English) Zbl 1027.05013
The authors introduce a new construction of central \((p^a,p^a,p^a,1)\) relative difference sets (which means that the forbidden subgroup is a central subgroup \(C\)) using orthogonal cocycles defined in terms of linearised permutation polynomials and finite presemifields. This construction yields a wealth of new non-abelian examples, while the case of commutative semifields reduces to a well-known previous construction. The cohomological approach is used to identify equivalence classes of central relative difference sets for which both \(C\) and \(G/C\) are elementary abelian. Several small cases are analyzed in detail.

MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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