Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver

Conjugacy and orthogonality of Latin squares have been widely studied in the literature not only for their theoretical interest in combinatorics, but also for their applications in distinct fields as experimental design, cryptography or code theory, amongst others. This paper deals with a series of binary constraints that characterize the sets of partial Latin squares of a given order for which their six conjugates either coincide or are all of them distinct and pairwise orthogonal. These constraints enable us to make use of a SAT solver to enumerate both sets. As an illustrative application, it is also exposed a method to construct totally symmetric partial Latin squares that gives rise, under certain conditions, to new families of Lie partial quasigroup rings

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$

Existence of perfect Mendelsohn designs with k=5 and λ>1

AbstractLet υ, k, and λ be positive integers. A (υ, k, λ)-Mendelsohn design (briefly (υ, k, λ)-MD) is a pair (X, B) where X is a υ-set (of points) and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair of points of X are consecutive in exactly λ blocks of B. A set of k distinct elements {a1, a2,…, ak} is said to be cyclically ordered by a1<a2<⋯<ak<a1 and the pair ai, ai+t is said to be t-apart in cyclic k-tuple (a1, a2,…, ak) where i+t is taken modulo k. It for all t=1,2,…, k-1, every ordered pair of points of X is t-apart in exactly λ blocks of B, then the (υ, k, λ)-MD is called a perfect design and is denoted briefly by (υ, k, λ)-PMD. In this paper, we shall be concerned mainly with the case where k=5 and λ>1. It will be shown that the necessary condition for the existence of a (υ, 5, λ)-PMD, namely, λv(υ-1)≡0 (mod 5), is also sufficient for λ>1 with the possible exception of pairs (υ, λ) where λ=5 and υ=18 and 28

Some Implications on Amorphic Association Schemes

AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition

Some Implications on Amorphic Association Schemes

AMS classifications: 05E30, 05B20;

A conjugate gradient algorithm for analysis of variance computations

Matrix oriented least squares or regression algorithms require substantial amounts of computer storage in order to solve analysis of variance problems. However, iterative methods exist which are capable of reducing the storage problem. These employ well-known balanced analysis of variance computations which do not require computer storage. The normal equations corresponding to a linear model with unbalanced data can be expressed in terms of the design matrix X for the cell means model. This fact can be used to construct algorithms which require a balanced analysis of variance problem to be solved in each iteration. A rule for constructing a generalized inverse of X\u27X which is positive definite and lower triangular is given. An iterative algorithm based on the modified conjugate gradient method to obtain the parameter estimates of an analysis of variance problem without storing X or X\u27X is developed using this inverse. This algorithm reduces the number of iterations required as compared to algorithms given previously. Further, the algorithm does not require reparameterization of the X matrix. An iterative method is also developed for calculating the sum of squares for testing a linear hypothesis in the original overparameterized model directly. Programs are implemented which compute the analysis of variance table and parameter estimates for linear models with unbalanced data using the above algorithms