Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146

Searching for incomplete self orthogonal latin squares : a targeted and parallel approach

The primary purpose of this dissertation is in the search for new methods in which to search for Incomplete Self Orthogonal Latin Squares. As such a full understanding of the structures involved must be examined, starting from basic Latin Squares. The structures will be explained and built upon in order to cover Mutually Orthogonal Latin Squares, Frame Latin Squares and Self Orthogonal Latin Squares. In addition the related structure Orthogonal Arrays, will be explained as they relate to Incomplete Self Orthogonal Latin Squares. This paper also dedicates time to explaining basic search methods and optimizations that can be done. The two search methods of focus are the backtracking algorithm and heuristic searches. In our 6nal method the two will work together to achieve an improved result. The methods currently being used to search in parallel are also provided, along with the necessary backup to there structure. The main research of this paper is focused on the search for Incomplete Self Orthogonal Squares. This is done by breaking down the problem into four separate areas of the square. By separating the blocks it enables us to work on a smaller problem while eliminating many incorrect solutions. The solution methodology is broken up into three steps and systematically solving the individual areas of the square. By taking advantage of the properties of squares to constrain our search as much as possible we succeeded in reducing the total search time significantly. Unfortunately, even with our improvement in the overall search time, no open incomplete self orthogonal latin square problems could be solved. Full results and comparisons to existing methods are provided