Variations on the McFarland and Spence constructions of difference sets

We give two new constructions of symmetric group divisible designs using divisible difference sets. These constructions generalize the McFarland and Spence constructions of symmetric designs. Divisible designs are of interest in several parts of mathematics: The design theorist needs classes of divisible designs as "ingredients" for recursive constructions of designs. Statisticians use divisible designs in the design of experiments. Recently, divisible designs found applications in cryptography, see [9]. A divisible design is an incidence structure consisting of m·n points partitioned into m classes of size n each. Two distinct points are joined by exactly Al or A,2 blocks depending whether the two points are elements in the same point class or not. The block size is a constant number and usually called k. The design is symmetric if the dual structure is a divisible design with the same' parameters. In many cases symmetric divisible designs are constructed as "developments" of divisible differ~nce sets. A divisible difference sets D in a group G is a k-subset of G with the following properties: The group G contains a subgroup N of order n and index m. The list of differences d-d' (d,d'e D) contains every nonzero element in N exactly Al times and the elements in G\N exactly A,2 times. We say that such a subset D is an (m,n,k,Al'A2)-divisible difference set in G relative to N. Using divisible difference sets it is quite easy to construct divisible designs. The points are just the elements of G and the blocks are the translates D+g={ d+g: de D} of D. Usually one requires that N is a normal subgroup of G. In this case one can show that the *Research partially supported by NSA grant MDA 904-92-H-3057, NSF grant NCR-9200265 and by an Alexander-von-Humboldt fellowship. The author thanks the Mathematisches Institut der Universitat Giessen for its hospitality during the time of this research. **Current address: Institut fUr Mathematik, UniversWit Augsburg, 86135 Augsburg, Germany AIlc::tV'~b<:i::!n IOllrn~1 of l.omhin<'ltorics lOC 1994). 00.199-204 l:Uln:::~pUllulllg Ul v l:HUlt: ut::Hgn 1:S :SYUllIlCllll:. nUWCVCI, U 1:S HUl KIIUWIl wnClIlcr any condition is at alL No example" of a non-symmetric divisible design is known that can be constructed from a group G with subgroup N as described above, see [2]. If A1=A2(=A) or n=l the divisible difference set becomes a (v,k,A)-difference set in the usual sense (v denotes the order of G) and the corresponding design is just a symmetric (v,k,A)-design. In this paper we want to describe a new construction of divisible difference sets (relative to a normal N) and therefore symmetric divisible As special cases we obtain difference sets due to McFarland [6], difference sets due to Spence [8], divisible difference sets due to Jungnickel (generalizing McFarland's construction) [3], divisible difference sets due to Jungnickel (generalizing Spence's construction) [ 4]. Informally, we can describe these four constructions as follows. Take the hyperplanes in V=GF(q)n and a group G. We construct divisible difference sets in VxG in the following way: Combine the elements of G with different hyperplanes or complements of hyperplanes of V. The constructions mentioned above and in the sequel differ just by the choice of the subsets S, T and U (in G) which we combine with hyperplanes, complements of hyperplanes and with the entire group V. We refer the reader to [2] for a source of many more constructions of divisible difference sets. Unfortunately, not many examples are known where A1=O, a situation which is of particular interest. Divisible difference sets with Al =0 are called relative difference sets. In order to describe our constructions and to verify that they yield divisible difference sets we need some notation. We will work in the group ring 7L G where G is written multiplicatively, and define CL agg)(t) := 2: aggt for integers t. A subset S of G will be identified with S:= 2:gE s g in the group ring 7L G and we denote this group ring element S, by abuse of notation. Using this notation we can translate the definition of a divisible difference set. A subset D of a group G is an (m,n,k,A1,A2)-divisible difference set in G relative to N if and only if D·D(-I) = k + Al (N-l) + A2(G-N) provided IGI=mn and INI=n. It is this group ring equation that we will check in the following constructions.