Definition of NBIBD (Nested Balanced Incomplete block Design)

NBIBD
Each block of a BIBD is split into sub-blocks of equal size, such that the sub-block design is also a BIBD. For example, a BIBD in blocks of 10 could give a new design with sub-blocks of 5 (and twice as many blocks as the original design). An alternative NBIBD would have sub-blocks of 2 (and five times as many blocks as the original design).
Informally, what this means is that each treatment appears in a block equally often with each other treatment, and likewise appears equally often (less often!) with every other treatment in a sub-block.
The notation adopted is NBIBD(v,b1,b2,r,k1,k2, λ1,λ2) or some shorter version thereof.

Designs of this type have been subject to much investigation. Furthermore, they are now researched under a variety of different titles, usually under the restriction that they be resolvable or near-resolvable, including (Generalised) Whist Designs Pitch Tournament Designs and Team Tournaments. There are also links from NBIBDs to Generalized Bhaskar-Rao designs: if a GBRD has every element of the group G equally often in each column, and if the non-zero elements of every pair of rows differ in the same number of places, then the result is equivalent to a NBIBD with g sub-blocks.

This definition of a NBIBD is due to Preece: a definition of a different sort of NBIBD is due to Federer. The two have been combined to produce a definition in which all blocks are partitioned (in the same way) into sub-blocks, not necessarily of the same size, and then a sub-set of the blocks of a given size in a main block are specified to belong to a BIBD. Thus if the block size were 20, and the partitioning were {2,2,2,3,3,4,4} then a design might be specified in which 3 sub-blocks of size 2 in each block were required to form a BIBD, as were the 2 blocks of size 3 were required to comprise a BIBD, while each of the blocks of size 4 would also comprise a BIBD.

Further conditions can be imposed, as in the designs discussed by Fuji_Hara et al.

One particular case of the former is where the sub-blocks are all of the same size: each of the sub-blocks is associated with a different BIBD. This is what Greig & Rees called a Type II nesting; the Preece nesting above was called a Type I nesting.

There are other forms of nested block design.