A Note on Optimal Partial Triallel Cross Designs

Triallel crosses form an important class of mating designs, which are used for studying the genetic properties of a set of inbred lines in plant breeding experiments. For p inbred lines, the number of different crosses for a complete triallel experiment is ( )( ) 3 3 1 2 2 p C p p p = − − of the type ( ) , i j k × 0,1, 2, , 1. i j k p ≠ ≠ = ... − Rawlings & Cockerham [5] were the first to introduce mating designs for triallel crosses. Triallel cross experiments are generally conducted using a completely randomized design (CRD) or a randomized complete block (RCB) design as environmental design involving 3 3 C crosses. Even with a moderate number of parents, say 10, p = in a triallel cross experiment; the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. For such situations, Hinkelmann [6] developed partial triallel crosses (PTC) involving only a sample of all possible crosses by establishing a correspondence between PTC and generalized partially balanced incomplete block designs (GPBIBD). Ponnuswamy & Srinivasan [7] and Subbrayan [8] obtained PTC using a class of balanced incomplete block (BIB) designs. Dharmlingum [9] also constructed PTC using Trojan squares. Actually Trojan squares are MOLS. Other research workers who contributed in this area are Arora & Aggarwal [10,11], Ceranka et al. [12]. More details on triallel cross experiments can be found in Hinkelmann [13] and Narain [14]. Das & Gupta [4] constructed block designs for triallel crosses by using nested balanced block design with parameters