In this paper we give an alternate proof that it is always possible to complete partial Latin squares with two filled rows and two filled columns, except for a few small counterexamples. The proof here is significantly shorter than the most recent proof by Adams, Bryant, and Buchanan. Additionally, we find sufficient conditions under which a partial Latin square with two filled rows and at leas...

We answer a question posed by Dénes and Keedwell that is equivalent to the following. For each order n what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order n? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal ...

Cycle switches are the simplest changes which can be used to alter latin squares, and as such have found many applications in the generation of latin squares. They also provide the simplest examples of latin interchanges or trades in latin square designs. In this paper we construct graphs in which the vertices are classes of latin squares. Edges arise from switching cycles to move from one clas...

Given a network of n = 2 gossipers, we want to schedule a cyclic calendar of meetings between all of them, such that: (1) each gossiper communicates (gossips) only once a day, with one other gossiper, (2) in every (n−1) consecutive days, each gossiper meets all other gossipers, and (3) every gossip, initiated by any gossiper, will reach all gossipers within k = log(n) days. In this paper we stu...

A partial Latin square (PLS) is a partial assignment of n symbols to an n×n grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a largest extension of a given PLS. In this paper we propose an efficient local search for this problem. We focus on the local search such that the neighborhood...

A Howell design of side s and order 2n, or more briefly an H(s, 2n), is an s x s array in which each cell is either empty or contains an unordered pair of elements from some (2n)-set V such that (1) every element of V occurs in precisely one cell of each row and each column, and (2) every unordered pair of elements from V is in at most one cell of the array. It follows immediately from the defi...

Selecting an optimal event distribution for experimental use in event-related fMRI studies can require the generation of large numbers of event sequences with characteristics hard to control. The use of known probability distributions offers the possibility to control event timing and constrain the search space for finding optimal event sequences. We investigated different probability distribut...

A Latin square is a complete assignment of [n] = {1, . . . , n} to an n × n grid such that, in each row and in each column, each value in [n] appears exactly once. A symmetric Latin square (SLS ) is a Latin square that is symmetric in the matrix sense. In what we call the constrained SLS construction (CSLSC ) problem, we are given a subset F of [n] and are asked to construct an SLS so that, whe...

In this paper, we propose an image encryption algorithm based on the chaos, substitution boxes, nonlinear transformation in galois field GF(2 8 ) and Latin square. Initially, the dynamic S-boxes are generated using Fisher-Yates shuffle method and piece-wise linear chaotic map. The algorithm utilizes advantages of keyed Latin square and affine-power-affine transformation to substitute highly cor...