On c‐Bhaskar Rao designs with block size 4

Several new families of c ‐Bhaskar Rao designs with block size 4 are constructed. The necessary conditions for the existence of a c ‐BRD (υ,4,λ) are that: (1)λ min =−λ/3 ≤ c ≤ λ and (2a) c ≡λ (mod 2), if υ > 4 or (2b) c ≡ λ (mod 4), if υ = 4 or (2c) c ≠ λ − 2, if υ = 5. It is proved that these conditions are necessary, and are sufficient for most pairs of c and λ; in particular, they are sufficient whenever λ− c ≠ 2 for c > 0 and whenever c − λ min ≠ 2 for c < 0. For c < 0, the necessary conditions are sufficient for υ> 101; for the classic Bhaskar Rao designs, i.e., c = 0, we show the necessary conditions are sufficient with the possible exception of 0‐BRD (υ,4,2)'s for υ≡ 4 (mod 6). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 361–386, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10009