Generalized relative difference sets and PBIBDs associated with amorphous association schemes over an extension ring of Z/4Z

Mieko Yamada Department of Applied Mathematics Konan University Okamoto, Higashinada-ku Kobe 658, Japan Partially balanced incomplete block designs, PBIBDs, are designs for which the property of balance of BIBD is relaxed. They are based on an association scheme. We introduce the concept of generalized relative difference sets, generalizing the concept of relative difference sets introduced by A.T. Butson[2] in 1963. We obtain a PBIBD from a generalized relative difference set translating by cyclic automorphisms. On the other hand, certain amorphous association schemes over the extension rings of Z/4Z were classified[6]. In this paper, we give a necessary and sufficient condition for the existence of generalized relative difference sets associated with these amorphous association schemes which give rise to PBIBDs, under some conditions. In the last section, we give examples of generalized relative difference sets associated with amorphous association schemes of class 3 for the case when the degree of an extension of Z/4Z is 3 and 4. 1 Generalized Relative Difference Sets and PBIBDs A.T. Butson[2] introduced the concept of relative difference sets in 1963. Relative difference sets are useful for construction of Hadamard matrices and D-optimal designs. See Spence[12,13,14] and Koukouvinos, Kounias and Seberry [9]. We recall the definition of relative difference sets. Definition. Let G be an additive abelian group of order v and D be a subset of G containing k elements. Let H be a subgroup of G of order h. If for d =I=0, dEG, the number of pairs (r, s) such that d = r s, r, s ED, has fixed values {~ when d(f..H, when dEH, then D is called a relative difference set. We extend this concept. Definition. Let G be an additive abelian group of order v and D be a subset of G containing k elements. Let Hi' H 2 , ... H t be subsets of G such that G = {O}UH1UH2U ... UH t , HinHj = 0, for i=l=-j. If for d 0, dEG, the number of pairs (r, s) such that d r, sED has fixed values when dEH 1 , when dEH 2 ,