Recursive Constructions of Balanced Incomplete Block Designs with Block Size of 7, 8 or 9

摘要

This paper gives constructions of balanced incomplete block designs and group divisible designs with k = 7, 8, or 9, and λ = 1. The first objective is to give constructions for all possible cases with the exception of 40, 78, and 157 values of v. Many of these initial exceptions have now been removed by Abel. In an update section, more are removed; group divisble designs with groups of size k(k - 1) are constructed for k = 7 and 8 with 124 and 87 exceptions; it is also established that v ≥ 294469 and v ≡ 7 mod 42 suffices for the existence of a resolvable balanced incomplete block design with k = 7. Group divisible designs with group size k and resolvable designs are constructed.