Perfect Graeco-Latin balanced incomplete block designs and related designs

TL;DR

This paper explores the construction of perfect Graeco-Latin balanced incomplete block designs and related designs, introducing new methods to generate optimal plans, including the first example of an optimal two-factor POTB that is not a PERGOLA.

Contribution

It provides direct and recursive construction methods for POTBs and presents the first known optimal two-factor POTB that is not a PERGOLA.

Findings

Constructed new classes of POTBs.

Presented the first optimal two-factor POTB not satisfying PERGOLA conditions.

Connected various combinatorial designs under a unified framework.

Abstract

Main effect plans orthogonal through the block factor (POTB) have been defined and a few series of them have been constructed in Bagchi (2010). These plans are very closely related to the `mutually orthogonal balanced nested row-column designs' of Morgan and Uddin (1996) and many other combinatorial designs in the literature with different names like `BIBDs for two sets of treatment', `Graeco-Latin designs' and `PERGOLAs'. In fact all of them may be viewed as POTBs satisfying one or more additional conditions, making them `optimal'. However, the PERGOLAs are defined to satisfy an additional property, without which also it is optimal. Interestingly, this additional property is satisfied by all the hitherto known examples of POTBs, even when their definitions do not demand it. In this paper we present direct and recursive constructions of POTBs. In the process we have constructed one design which seems to be the first example of an `optimal' two-factor POTB which is not a PERGOLA (see Theorem \ref {POTB2}).