Cavenagh, Nicholas J.Nie, Xiao2025-01-302025-01-302024-09-23https://hdl.handle.net/10289/17143A row-column design is any rearrangement of the blocks of a combinatorial design into a rectangular array. A row-column block design is a row-column design in which the blocks form a balanced incomplete block design; that is, each pair of elements occurs in a constant number of blocks. In this thesis we study the efficiency and structure of row-column block designs. In particular, we use solutions to Heffter’s difference problem to give construct row-column block designs with 3 elements per cell with optimal regularity in rows and columns. In Section 1 we review definitions and theorems related to Latin squares. We introduce related concepts of balanced incomplete block designs and incidence matrix and concurrence matrix of designs. In Section 2 we give our main results of row-column block designs with block size 3. In Section 3 we use Heffter’s difference problem to give some solutions. In Section 4 we explain efficiency measures for block designs. In Section 5 we introduce Trojan semi-Latin squares. We give efficiency measures for Trojan semi-Latin squares. In Section 6 we show the applications to experimental design. Original results are given in the following theorems: Theorem 2.9, Theorem 3.1, Theorem 3.3, Theorem 3.4, Theorem 4.2, Theorem 5.2, Theorem 5.12. Results from Section 2 and Section 3 have been submitted as a manuscript for publication [17] (Xiao Nie, Row-column block designs with blocks of size three, Ars Combinatoria, submitted).enAll items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.Latin Squaresorthogonal Latin squaremutually orthogonal Latin squaresemi-Latin squareuniform semi-Latin squarerow-column designrow-column block designbalanced incomplete block design (BIBD)scaled information matrixThesis