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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 .

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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 $\mathsf{ BIBD}?$

Sufficiency is a tough job to show.

Check this paper by Haim Hanani for further insight. Also, go through chapter $12$ of $\rm [I].$

Reference:

$\rm [I]$ Constructions and Combinatorial Problems in Design of Experiments, Damaraju Raghavarao, Dover Publications, $1988.$

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  • $\begingroup$ No , it's not possible that every design formed by these parameters will be a BIBD , that is what is confusing me , I read a question in exam to state and prove sufficient condition but what to write if there doesn't exist a sufficient condition $\endgroup$
    – simran
    Commented Dec 14, 2022 at 3:20

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