# Atomic Latin Squares based on Cyclotomic Orthomorphisms

Atomic latin squares have indivisible structure which mimics that of
the cyclic groups of prime order. They are related to perfect
1-factorisations of complete bipartite graphs. Only one example of an
atomic latin square of a composite order (namely 27) was previously
known. We show that this one example can be generated by an
established method of constructing latin squares using cyclotomic
orthomorphisms in finite fields. The same method is used in this paper
to construct atomic latin squares of composite orders 25, 49, 121,
125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It
is also used to construct many new atomic latin squares of prime order
and perfect 1-factorisations of the complete graph K_{q+1} for many
prime powers q. As a result, existence of such a factorisation is
shown for the first time for q in
{529,2809,4489,6889,11449,11881,15625,22201,24389,24649,
26569,50653,78125,79507,103823,161051,205379,300763,
357911,371293,493039,571787}
We show that latin squares built by the `orthomorphism method' have
large automorphism groups and discuss conditions under which different
orthomorphisms produce isomorphic latin squares. We also introduce an
invariant called the train of a latin square, which proves to be
useful for distinguishing non-isomorphic examples.
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