Nested balanced ternary designs and Bhaskar Rao designs

In this paper, we consider balanced ternary designs, BTDs, in which every block contains one element singly and the rest doubly. We call these packed BTDs, and we investigate three aspects of these designs: existence, nestings and signings. Construction methods generate classes of packed BTDs that are nested with balanced (BIBD) or partially balanced (PBIBD) incomplete block designs. Some of these classes are signed to produce c-Bhaskar Rao BTDs, most often with c = 0. Packed BTDs with block size three and five are studied in detail. The spectrum of possible indices for packed BTDs is determined. In particular, we prove every triple system with index 3t is nested within a packed Bhaskar Rao balanced ternary design with K = 5 and index 8t. We give several new families of PBIBDs for block size 3, and show that each is nested within a BTD with block size 5 and index 4 or 6. We show that the necessary conditions are sufficient for the existence of this type of BTD, whether or not it has a design nested within.