# Necessary and sufficient condition for existence of balanced incomplete block design

State and prove the necessary and sufficient condition for an incomplete block design to be balanced

A balanced incomplete block design (BIBD) is an incomplete block design in which

• b blocks have the same number k of plots each and
• every treatment is replicated r times in the design.
• Each treatment occurs at most once in a block, The three conditions of balancedness are
1. $$rv = bk$$

2. $$\lambda(v-1) = r(k-1)$$

3. $$r > k$$

Where r is number of repetitions, v is number of treatments , b is number of blocks , $$\lambda$$ is number of pairs of treatments , k is block size .

Here the problem is that these are only the necessary conditions , I have been searching in various books but could not find the Sufficient condition for balanced ibd .

Are they sufficient? Can you always frame a $$\sf BIBD$$ with all those sets of parameters that satisfy the conditions stated? Does there exist any counter example? Do certain conditions other than those exist that are necessary as well as sufficient that encompass all forms of $$\mathsf{ BIBD}?$$
Check this paper by Haim Hanani for further insight. Also, go through chapter $$12$$ of $$\rm [I].$$
$$\rm [I]$$ Constructions and Combinatorial Problems in Design of Experiments, Damaraju Raghavarao, Dover Publications, $$1988.$$