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Atomic Latin Squares based on Cyclotomic Orthomorphisms

Wanless, IM (2005). Atomic Latin Squares based on Cyclotomic Orthomorphisms. Electronic Journal of Combinatorics,12(R22):1-23.

Document type: Journal Article
Citation counts: Scopus Citation Count Cited 10 times in Scopus Article | Citations
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Title Atomic Latin Squares based on Cyclotomic Orthomorphisms
Author Wanless, IM
Journal Name Electronic Journal of Combinatorics
Publication Date 2005
Volume Number 12
Issue Number R22
ISSN 1077-8926   (check CDU catalogue open catalogue search in new window)
Scopus ID 2-s2.0-21244476688
Start Page 1
End Page 23
Total Pages 23
Place of Publication 2005
Publisher Elsevier
HERDC Category C1 - Journal Article (DEST)
Abstract 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 Kq+1 for many prime powers q. As a result, existence of such a factorisation is shown for the first time for q in {529,2809,4489,6889,11449,11881,15625,22201,24389,24649,26569,29929,32041,38809,44521,50653,51529,52441,63001,72361,76729,78125,79507,103823,148877,161051,205379,226981,300763,357911,371293,493039,571787}. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we 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.
Keywords factorizations
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