On designs (22,33,12,8,4) (Q1112826)

This paper deals with the existence of balanced incomplete block designs (BIB) with parameters \(\nu =22\), \(b=33\), \(r=12\), \(k=8\), \(\lambda =4\). Using the results due to John Van Rees obtained through personal communication, the authors enumerate up to isomorphism all \(C_ 5's\) where \(C_ 5\) is a set of 5 columns in the incidence matrix in which every row has an even number of 1's. Of the 108 cases enumerated, all but 31 are rejected. The last section reduces these 31 cases to 13 only. Thus the existence of a (22,33,12,8,4) BIB design has been reduced to a study of 13 five column starts. Thus more needs to be done before this problem is finally resolved.