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The study of building sets through product and lifted constructions. (English) Zbl 1208.05009

Summary: The studies of difference sets are important in design theory because a \((v,k,\lambda)\)-difference set in a group \(G\) is equivalent to a symmetric \((v,k,\lambda)\)-design with a regular automorphism group \(G\).
In this paper, a unifying construction for difference sets, Davis and Jedwab present a recursive construction for difference sets, which unifies the Hadamard, McFarland and Spence families. The construction also yields a new family of difference set which is known as Davis and Jedwab difference set. One of the essential ingredients in their construction is building set. Davis and Jedwab are showed that the existence of building sets implies the existence of families of semi-regular relative difference sets or divisible difference sets.
In this paper, we study two methods to construct building sets, which are, lifted construction and product construction. By using the lifted construction, we obtained \((pta,m,1)\)-BS on \(G\times\langle w|w^p= 1\rangle\times \langle h|h'= 1\rangle\) relative to \(U\) provided there is an \((a,m,pt)\)-BS on \(G\) relative to \(U\). On the other hand, by applying the product construction, we obtained building sets in Abelian groups of order \(p^{2(k+2)r}\) relative to a subgroup of order \(p^r\), where \(r\) and \(d\) are integers such that \(r> 0\) and \(d\geq 0\).
Finally, we also classify some families of building sets in Abelian 2-groups.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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