A large set of disjoint S(*; t, k, v) designs, denoted by LS(*; t, k, v), is a partition of k-subsets of a v-set into S(*; t, k, v) designs. In this paper, we develop some recursive methods to construct large sets of t-designs. As an application, we construct infinite families of large sets of t-designs for all t. In particular, we show that if v=2 t&3 m&2, k=2 t&3 &1, and t, m 2, then a LS(( v&t k&t )Γ2; t, k, v) exists.
1996 Academic Press, Inc.
1. Introduction
Let v, k, t, and * be four positive integers such that v k t>0. We denote the set of i-subsets of a set X by P i (X).
A t-design S(*; t, k, v) is a pair (X, B) in which X is a finite set with cardinality v and B is a collection of elements of P k (X) such that every element of P t (X) appears exactly * times in B.
A large set of disjoint S(*; t, k, v) designs, denoted by LS(*; t, k, v), is a partition of the k-subsets of a v-set into S(*; t, k, v) designs. Obviously, if a LS(*; t, k, v) exists, then *=( v&t k&t )Γn for some n. For the sake of simplicity, we write LS(1Γn; t, k, v) instead of LS(( v&t k&t )Γn; t, k, v). A well known necessary condition for the existence of a LS(1Γn; t, k, v) is that n|( v&i k&i ) for i=0, ..., t. It is well known that in each of the following cases these conditions are also sufficient: (i) t=1, [2] (ii) t=2, k=3 and v{7, [3,[7][8][9][10][11][13][14][15] (iii) t=2, k 15 and n=2, [1, 4] (iv) t=3, k=4 and v#0 (mod 3), [13] (v) t=6, k=7 and n=2, [5,6,12] (vi) v=nl(t)+t and k=t+1 in which l(t) is defined recursively by l(i)=l(i&1) l.c.m.[1, ..., i+1] l.c.m.[( i j )| j=1, ..., i], l(0)=1 [12]. For t>6, designs of the last class together with their derived and residual designs are the only known designs.