Diagonally Cyclic Latin Squares

A latin square of order n possessing a cyclic automorphism of order n is said to be diagonally cyclic because its entries occur in cyclic order down each broken diagonal. More generally, we consider squares possessing any cyclic automorphism. Such squares will be named after Parker, in recognition of his seminal contribution to the study of orthogonal latin squares. Our primary aim is to survey the multitude of applications of Parker squares and to collect the basic results on them together in a single location. We mention connections with orthomorphisms and near-orthomorphisms of the cyclic group as well as with starters, even starters, atomic squares, Knut-Vik designs, bachelor squares and pairing squares.

In addition to presenting the basic theory we prove a number of original results. The deepest of these concern sets of mutually orthogonal Parker squares and their interpretation in terms of orthogonal arrays. In particular we study the effect of the various transformations of these orthogonal arrays which were introduced by Owens and Preece.

Finally, we exhibit a new application for diagonally cyclic squares; namely, the production of subsquare free squares (so called N squares). An explicit construction is given for a latin square of any odd order. The square is conjectured to be N and this has been confirmed up to order 10000 by computer. This represents the first published construction of an N square for orders 729, 2187 and 6561.

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