The Anatomy of a Design

摘要

We give a proof of a family of 2t-associate class partially balanced designs with two distinct indices, γ_1 = 2 and γ_2 = 1, denoted {PBIBD(4t+l,3;2,l): t ≥1). Moreover, we wish to provide an exposition of our solution to a concrete problem: does there exist a partially balanced incomplete block design on 93 points with block size 3 and with only two distinct indices, a PBIBD(93,3;2,1)? This specific problem led to an array of new designs and, in many cases, a partition of a BIBD(4t+1, 3, 3) into two such PBIBDs. Along the way we touch on the topics, in no particular order, of BIBDs, large sets, cyclic designs, nested designs, ternary designs, experimentation and visualization with Excel, complimentary families of difference sets, and Skolem sequences.