Balanced Semi-Latin Rectangles: Properties, Existence and Constructions for Block Size Two

Abstract

There exists a set of designs which form a subclass of semi-Latin rectangles. These designs, besides being semi-Latin rectangles, exhibit an additional property of balance, where no two distinct pairs of symbols (treatments) differ in their concurrences, that is, each pair of distinct treatments concurs a constant number of times in the design. Such a design exists for a limited set of parameter combinations. We designate it a balanced semi-Latin rectangle and give some properties and necessary conditions for its existence. Furthermore, algorithms for constructing the design for experimental situations where there are two treatments in each row-column intersection (block) are also given. KeywordsBalanced incomplete block design (BIBD) • Balanced semi-Latin rectangle (BSLR) • Optimal design • Quotient block design (QBD) • Regular graph design (RGD)

Constructions for regular-graph semi-Latin rectangles with block size two

“…Row-column designs such as Latin squares and semi-Latin squares can be seen to be special cases of the SLR with n r = n c = 1: see Uto and Bailey (2020). Discussions on semi-Latin squares can be found in papers such as Preece and Freeman (1983), Bailey (1988Bailey ( , 1992, Bailey and Chigbu (1997), Bailey and Royle (1997), Bedford and Whitaker (2001), Soicher (2012), Soicher (2013) and Bailey and Soicher (2021).…”

“…SLRs whose quotient block designs are BIBDs are named balanced semi-Latin rectangles (BSLRs) by Uto and Bailey (2020). That paper examined the combinatorial properties and necessary conditions for a BSLR to exist, and gave algorithms for constructing these designs when the block size is two.…”

“…Hence ∆ is a RGSLR. □ For the case that k = 2, Uto and Bailey (2020) give BSLRs. Theorem 2 shows that these can be adjoined to RGSLRs of suitable size constructed in Sections 2-5 to obtain larger RGSLRs.…”

“…Example 7. Figure 4 of Uto and Bailey (2020) shows a (3 × 15)/2 BSLR for six treatments. This can be adjoined to the (3 × 12)/2 RGSLR in Fig.…”