Construction methods for row-complete Latin squares

Abstract

A latin square L of order n is said to be row-complete, and is denoted by RCLS(n), if the ordered pairs (Lij, Lij+1) are all distinct for 1< i < n and 1 < j < n -1. Row-complete latin squares are also called roman squares, and are used in statistics, an RCLS(n) is a balanced repeated measurements (n,n,n) design.

In 1949, Williams provided a simple construction for an RCLS (2m) for every m, but the situation for odd orders has proven to be much more difficult. In the last 30 years or so, various authors have given constructions of RCLS for certain odd orders, but the state of knowledge has nevertheless remained somewhat sparse.

In this thesis, two new methods of construction for RCLS are given. The first, a product construction, yields infinitely many new orders for which RCLS are known to exist. The second construction, which is the highlight of this thesis, is a direct construction of an RCLS for any odd composite order other than 9. Since RCLS of order 9 and of even order have previously been constructed, this proves that RCLS of every composite order exist.

In addition, a new result is given on the related concept of quasi-complete latin squares (QCLS). Specifically, it is shown that complete sets of mutually orthogonal QCLS(p) exist for every prime p. Such sets were previously known to exist only for primes p < 13.