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