A Holey NBIBD, a (k2, k1)HNBIBD(v,G), is a design for v treatments in main blocks of size k1, and sub-blocks of size k2, in which the treatment sets are partitioned into groups defined in G, such that a pair of treatments from the same group occur together in the same block neither in the main blocks, nor in the sub-blocks (analogous to a GDD). All other pairs of treatments occur together in main and sub-blocks λ1 and λ2 times, respectively. This is usually referred to in the form (k2, k1)HNBIBD(v) of type G.
A (k2, k1)HNBIBD(v) with group type 1(v-h)h1 will also be called an Incomplete NBIBD denoted by (k2, k1)INBIBD(v,h). Note that, for h=0 or h=1, every (k2, k1)NBIBD(v) can be considered as a (k2, k1)INBIBD(v,h).
Here are some examples of INBIBDs
(2,6)INBIBD(11,2): (∞0, 22 | 00, 11 | 12, 21) (∞1, 02 | 00, 01 | 10, 20) mod (3,3)
(2,6)INBIBD(16,3): (∞0, 7 | 0, 4 | 6, 9) (∞1, 10 | 0, 2 | 3, 11) (∞2, 8 | 0, 1 | 5, 12) mod 13
(3,6)INBIBD(11,2): (∞0, 22, 00 | 11, 12, 21) (∞1, 02, 00 | 01, 10, 20) mod(3,3)
(3,6)INBIBD(16,3): (∞0, 2, 3 | 0, 4, 11) (∞1,7, 12 | 0, 6, 9) (∞2, 0, 1 | 5, 8, 10) mod 13
The ingredients for many more INBIBD's and HNBIBD's with a main block size of 6 may be obtained here; see here for an explanation of how to get from a PMD to a NBIBD.
A general theorem for constructing some useful INBIBDs can also be obtained by conversion from a theorem for IOBIBDs given here.
If the invariants are ignored, then these designs can be regarded as examples of more general nested designs: for example, the (3,6)INBIBD(16,3) would be a BIBD(13,5,5) with a PBD(13,{2,3},2) nested within, while a (3,6)INBIBD(27,4) becomes a PBD(23,{5,6},5) with a PBD(23,{2,3},2) nested within.
An Incomplete HNBIBD, an IHNBIBD, is a HNBIBD with an additional hole and is analogous to an IGDD. The type of an IHNBIBD is written similarly in exponential notation, as for example, {(m1,n1)h1, (m2,n2)h2,...}.
If there exists a PBD(v,K,1), and for each n ∈ K here exists a (k2, k1)NBIBD(n), then there exists a (k2, k1)NBIBD(v).
If a (k2, k1)INBIBD(v,h) exists, and there also exists a (k2, k1)NBIBD(h), then a (k2, k1)NBIBD(v) exists.
If there exist a (k2, k1)HNBIBD(v) with group type vector of (h1, h2,...,hu), a (k2, k1)INBIBD(hi+w,w), for each i, 1≤i≤ (u -1), and a (k2, k1)NBIBD(hu + w), then there exists a (k2, k1)NBIBD(v + w). Note that w may be zero.
Suppose there exists a ``master'' GDD (X,G,B) with index λ' and that w is a positive function on X. Suppose also that there exists a ``subordinate'' (k2,k1)HNBIBD of type {w(x):x ∈ B} for every B ∈ B. Then there exists a (k2,k1)HNBIBD of type {{∑ w(x): x ∈ G} for G ∈ G}, on v vertices, where v = {∑ x : x ∈ G, G ∈ G}.
Suppose there exists a ``master'' (k2,k1)HNBIBD of type (v1, v2,...,vh) Suppose also that there exists a ``subordinate'' TD(k1,m). Then there exists a (k2,k1)HNBIBD of type (mv1, mv2,...,mvh).
Let (X,G,B) be a (k2, k1)HNBIBD(v), with group partition H. Let F be a set of new points, and suppose that for each group Gi ∈ G, there exists a (k2, k1)HNBIBD(hi) given by (Gi ∪ F, HGi∪{F}, Bi) ,where HGi={Hi,1,Hi,2,...,Hi,j,...} and Gi=∪{Hi,j}, Bi being the blocks. Then there exists a (k2, k1)HNBIBD with treatment set X ∪ F, groups F∪{∪HGi : Gi ∈ G}, and blocks B ∪{∪ Bi : Gi ∈ G}.
If there exist:
then there exists a (k2, k1)INBIBD(vm+d,vn+d). Additionally, if either there exists a (k2, k1)NBIBD(vn+d), or there exist both a (k2, k1)INBIBD(vn+d,n+d) and a (k2, k1)NBIBD(m+d), then there exists a (k2, k1)NBIBD(vm+d).
If there exist
then there exists a (k2, k1)INBIBD(vm+d,x) for x = d or x=(m + d). If there exists also a (k2, k1)NBIBD(x) then there exists a (k2, k1)NBIBD(vm+d)).
If there exists
then a (k2, k1)NBIBD(vm) exists. Moreover, this NBIBD contains a (k2, k1)NBIBD(v) as a subdesign, so a (k2, k1)INBIBD(vm,v) also exists.
In practice, some standard constructions or types of PBDs (with λ=1) are then used in a fairly standard way as listed below.
A BIBD is a PBD, so any BIBD with λ = 1 can be used with the construction inflating a PBD given above.
To a RBIBD with v points, replication r,block size k and λ=1, can be added s points, where 1 ≤ s ≤ r. Each new point is associated uniqely with a replication, and is added to each of the blocks of that replication. A new block is added to the design, consisting exclusively of the new points. This is a PBD for v+s points with block sizes k, k+1 and s.
A TD(k, m) is a PBD for km points and with block sizes k and m.
Suppose a TD(k+1,m). From one of the groups, delete all but s points. The result is a PBD with for km+s points and with block sizes k,k+1,m and s.
Suppose a TD(k+s,m). Keeping one block intact, from s of the groups delete all but one of the points. The result is a PBD for km + s points and with block sizes k, k+1, k+s and m.