Construction and comparison of semi-Latin rectangles

Abstract

This work is concerned with semi-Latin rectangles (SLRs). These designs are row-column designs with nice combinatorial properties; and were introduced in Bailey and Monod (2001). They generalize the Latin squares (LSs) and semi-Latin squares (SLSs) and are useful for many experimental situations in diverse sectors, ranging from agriculture to the industry. We classify these designs as balanced semi-Latin rectangles (BSLRs) and non- balanced semi-Latin rectangles (NBSLRs) and develop some constructions, via algorithms, for good SLRs, that is, SLRs with good statistical properties for each classification using some combinatorial approaches. BSLRs do not always exist, but when they exist, they are optimal among other SLRs in their class over a range of criteria. When a BSLR does not exist, good designs can be sought among RGSLRs, particularly for large number of blocks, if they exist. Hence for the NBSLRs we concentrate on regular-graph semi-Latin rectangles (RGSLRs). For each classification, constructions are given for designs with block size two and for those with block size larger than two; and for block size two, we consider situations when the number of treatments is odd and also when it is even. The construction involving RGSLRs with block size two having an odd number of treatments is generalized to accommodate more columns and a table showing starters in some cyclic groups of small odd orders, 5 to 15 is given to facilitate the construction. Some direct constructions, for different situations, have been developed for RGSLRs whose number of treatments is even and whose block size is two less the number of treatments. These are backed up with some examples, which when compared with designs of the same size obtained via complementation, they are found to be identical under one of the methods but isomorphic under the other method. Finally, for each of BSLRs and RGSLRs, we have given a table containing sets of parameters, which can combine to give a design alongside their construction and also where the design (or its construction, as the case may be) can be found in the thesis.