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Báo cáo toán học: "Atomic Latin Squares based on Cyclotomic Orthomorphisms"
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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Atomic Latin Squares based on Cyclotomic Orthomorphisms. | Atomic Latin Squares based on Cyclotomic Orthomorphisms Ian M. Wanless School of Engineering and Logistics Charles Darwin University NT 0909 Australia ian.wanless@cdu.edu.au Submitted Feb 19 2005 Accepted May 2 2005 Published May 9 2005 Mathematics Subject Classifications 05B15 05C70 11T22 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. This work was undertaken at Christ Church Oxford and at the Department of Computer Science Australian National University. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R22 1 1 Introduction Group theorists think of the cyclic groups of prime order as their basic building blocks. Every Cayley table of a hnite group is a latin square and the .