A well-known construction of BIBDs uses doubly homogeneous groups: a basic block is obtained which is equivalent to a sub-group of the multiplicative subgroup of the d.h. group. The orbit of this block under the d.h. group is a BIBD with r = (v-1)/2.
Evidently, if such a basic block were itself partitioned into sub-blocks of equal size, then the resulting design would be an NBIBD with r = (v-1)/2.
There are 5 sets of designs shown which use this method: see the menu alongside. One design is missing, for v=343, which is a prime power, not a prime number - I'll write that up as a separate page some time.
A similar result applies when a sub-group of the additive sub-group of a d.h. group is used - when v is a non-trivial odd power of a prime of the form 4t+3. Start off with a Galois Field of order v. The main block is partitioned into the cosets (with respect to itself) of a sub-sub-group of the order of the sub-blocks. The initial blocks are the cosets (with respect to the Galois Field) of the main block thus constructed. The full design is obtained by multiplying these initial blocks by the even powers of the primitive root. It is easier to take the discrete logarithms of the elements of these initial blocks and to cycle additively, mod (v-1), in steps of 2. Some solutions are listedhere, while an applet to provide solutions ishere.
Nearly all the NBIBDs published to date have parameter sets the same as designs produced in these ways using doubly transitive groups (in particular, r = (v-1)) so most of the designs presented here are new, so far as I can ascertain. One design which has been noted previously (in Abel, Finizio, et al, for example) is for 27 treatments in blocks of 9 and 3: possibly this is one of a series with v = 36t+27, but, as these values are not prime, solutions will be harder to find.