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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