Some new D-optimal designs

We construct several new (v; r, 8; A) supplementary difference sets with v odd and T' + .5 = A + (v 1) /2. They give rise to D-optimal designs of order 2v. D-optimal designs of orders 158, 194, and 290 are constructed here for the first time. We also give an up to date survey of this class of supplementary difference sets in arbitrary Abelian groups of odd order v < 100. o. Introduction Supplementary difference sets (SDS) in finite Abelian groups is an active topic of research. Examples of supplementary difference sets were given as early as 19:39 in a paper of Bose [1]. More recently, these have been formally defined and popularized in the work of J. Seberry (Wallis) [1.5, 17, 18, 19]. In this paper we consider only one special class of supplementary difference sets (X, Y) in a finite Abelian group G of order v. This means that every a E G, a #0, can be represented as a = x y with x, y E X or x, y E Y in A ways (in total), where A is a constant independent of a. If IXI = T' and IY\ = 8, we say that (X, Y) have parameters (v; r, 8; A). Furthermore we require that v be odd and r+8 = A+( v-I) /2. Such supplementary difference sets will be called D-optimal because they give rise to D-optimal designs of order 2v. D-optimal SDS's have been studied by many authors starting with Ehlich [9]. There is only one infinite series of such sets known at the present time (see [11]). We present several new D-optimal SDS's with v = 27,49,73,79,97,11:3, and 14.5. In particular, we construct for the first time D-optimal designs of orders 1.58, 194, *Supported in part by the NSERC Grant A-5285. Australasian Journal of Combinatorics 15(1997), pp.221-231 and 290. For v = 27,49 our SDS's lie in an elementary Abelian group. No such SDS's were known before for these values of v. For v = 73 the known SDS has parameters (73; 36, 28; 28) and we construct three non-equivalent SDS's having a different set of parameters, namely (73; 42, 30; 36). For v = 113 a D-optimal design of order 2v is known, it belongs to an infinite series constructed by A.L. Whiteman [20]. His construction does not use SDS's. The first example of SDS with parameters (113; 49, 49; 42) was found recently (see [12]). Our SDS with the same parameters is not equivalent to that example. We collect in Table 1 all known results about the existence of D-optimal SDS's, with parameters (v; r, s; .\), v < 100, that satisfy the known necessary conditions. There are a number of undecided cases indicated by the question mark. 1. D-optimal Supplementary Difference Sets Let G be a finite Abelian group (written multiplicatively) of order v, and ZG its group ring over Z (the ring of integers). For X C G let X' G\X denote the complement of X in G, and let N(X) = (LX) . (L xl ) E ZG. xEX xEX We also set T = LX. xEG We say that the ordered k-tuple (Xl, ... ,Xk ), with Xi c G, are supplementary difference sets (SDS) with parameters (v; nl,···, nk; A) if IXil = ni for i = 1,· .. ,k, and ~N(X;) = (~ni -A) .1+ AT, where 1 EGis the identity element. For k = 1 we obtain the definition of difference sets. We now introduce a special class of SDS. Definition (1.1) We say that (v; r, S; A) supplementary difference sets are D-optimal if v is odd and r + s = A + (v 1) /2. We shall see in the next section why these SDS's are important. The following two propositions are well known. Proposition (1.2) If (X, Y) are D-optimal SDS, then the same is true for (X', Y), (X, yl), and (X', Y'). III Proposition (1.3) If (X, Y) are D-optimal SDS with parameters (v; r, S; A), then 2(2v 1) = (v 2r)2 + (v 2S)2. (1.1)