Nested Balanced Incomplete Block Designs of the Series I and II Type
Suppose there exists a Balanced Incomplete Block (BIB) design (v, b, r, k, ) for which an initial block solution could be obtained from t-initial blocks. Equally if it is possible to divide each initial block into m sub-blocks from the initial block for generating a BIB design with v treatments arranged into blocks of size k2. Then by systematic development of these initial blocks, we have NBIB designs for both the series II and I. Designs constructed in this paper, which are referred to as of series-I and series-II, are indeed of the Nested Balanced Incomplete Block Designs (NBIBD's) type and they do exist for some values of v. Meanwhile, NBIBD's for series-I exist for all odd values of v, while NBIBD's for series-II equally exist for all even values of v. Also if sequence of the treatment (v) for series-II is one fewer than series-I, both series-I and series-II give the same initial block and consequently the main block of size k1 for series-I is equal to k1 for series-II, and correspondingly the size of the sub-blocks, k2 for series-I is equal to k2 for series-II. Other design parameters exhibit similar interesting symmetry.