Some Relations on Paratopisms and An Intuitive Interpretation on the Adjugates of a Latin Square

This paper will present some intuitive interpretation of the adjugate transformations of arbitrary Latin square. With this trick, we can generate the adjugates of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and adjugate transformations in composition will also be shown. It will solve the problem that when F1*I1=I2*F2 how can we obtain I2 and F2 from I1 and F1, where I1 and I2 are isotopisms while F1 and F2 are adjugate transformations and * is the composition. These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares.Comment: Any comments and criticise are appreciate

An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).Comment: 19 pages, 10 figure

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