Mutually nearly orthogonal Latin squares and their applications.

摘要

This work includes both statistical and combinatorial results for mutually nearly orthogonal Latin squares (MNOLS) of order v = 2 m. MNOLS are closely related to mutually orthogonal Latin squares (MOLS); MNOLS modifies the orthogonality condition of MOLS slightly.; Experiments are considered for main effects plans where we use 2 and 3 factors. In both cases results are given in general and for the specific orders 6 and 10. In these situations it is not possible to use MOLS since two of order 6 do not exist, and three of order 10 are not known to exist.; We also consider a carryover effect plan with 2 factors, where one has a carryover effect of lag 1. The optimality of the Williams design makes it a logical choice for the factor with carryover effects. However, as is shown in this work, the Williams design is orthogonally isolated. Therefore the analysis here considers a nearly orthogonal Latin square to the Williams design for the second factor with direct effects only. Relative efficiencies are given for several orders. In this case we present some nice mathematical results for the orders v = 2m, m odd.; On the combinatorial side of things, we show that a maximal set of MNOLS of order 6 does not exist. Two constructions for 2 MNOLS of any even order are given and 3 MNOLS of several orders are provided.; As MOLS are extended to F-squares, we modify MNOLS and introduce mutually nearly orthogonal F-squares (MNOFS). A bound is given for the number of MNOFS, as well as a maximal set for a certain order.