Balanced Incomplete Block Designs with Block Size 8

The necessary conditions for the existence of a balanced incomplete block design on v greater than or equal k points, with index λ and block size k, are that: λ(v - 1) equivalent 0 mod (k - 1) λv(v - 1) equivalent 0 mod k(k - 1) For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is v = 3, 753. For these 38 values of v, we show (v, 8, λ) BIBDs exist whenever λ > 1 for all but five possible values of v, the largest of which is v = 1, 177, and these five v's are the only values for which more than one value of λ is open. For λ > 1, we show the necessary conditions are sufficient with the definite exception of two further values of v, and the possible exception of 7 further values of v, the largest of which is v = 589. In particular, we show the necessary conditions are sufficient for all λ > 5 and for λ = 4 when v does not equal 22. We also look at (8, λ) GDDs of type 7 m. Our grouplet divisible design construction is also refined, and we construct and exploit α-frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n : n equivalent 0, 1 mod 8, n greater than or equal 8}, and construct a few new TDs with index > 1. © 2001 John Wiley and Sons, Inc.