User:Melapolis/Group Divisible Designs

Group Divisible Designs are a type of Combinatorial design .

Formal Definition
Let v>2 be a positive integer. A group divisible design or GDD is a triple (X,G,A) such that the following properties are satisfied: We note that groups of size one are allowed
 * X is a finite set of elements called points
 * G is a partition of X into at least two nonempty subsets called groups or holes
 * A is a set of subsets of X called blocks such that |A| is greater than or equal to 2 for every block of A
 * A group and a point contain at most one common point
 * Every pair of points from distinct groups is contained in exactly one block

Theorems

 * Suppose that v>k>1. Then there exists a (v,k,1)- BIBD if and only if there exists a group divisible design having v-1 points, r groups of size k-1 and blocks of size k where r=(v-1)(k-1).