The Existence Of (3, λ)-GDDs Of Type G~tw~1

Letλbe a positive integer. A group divisible design (GDD) of indexλis a triple (X, (?), B), where X is a finite set of points, (?) is a partition of X into subsets called groups, B is a collection of subsets of X (called blocks), such that a group and a block contain at most one common point, and every pair of points from distinct groups occurs in exactlyλblocks. Group divisible design is one of basic designs in the theory of combinatorial design, and it plays an important role in the process to construct the other kinds of designs. Whenλ= 1, the existence of 3-GDD of type g~tw~1 has been solved by Colbourn, Hoffman and Rolf Rees. From this, the article will determine the existence for the caseλ≥2, combined with the result of the caseλ= 1, and we can completely solve the existence of (3,λ)-GDDs of type g~tw~1, that is:Let g, t, w andλbe nonnegative integers. The necessary and sufficient conditions for the existence of a (3,λ)-GDD of type g~tw~1 are that(i) if g＞0, then t≥3; or t=2 and w=g; or t=1 and w=0; or t=0;(ii) w≤g(t－1) or gt=0;(iii)λ(g(t－1)＋w)≡0 (mod 2) or gt=0;(iv)λgt≡0 (mod 2) or w=0; and(v)λ(1/2g~2t(t－1)＋gtw)≡0 (mod 3).