Frame Self-orthogonal Mendelsohn Triple Systems |
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Authors: | Email author" target="_blank">Yun?Qing?XuEmail author Han?Tao?Zhang |
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Institution: | (1) Department of Mathematics, Northern Jiaotong University, Beijing 100044, P. R. China;(2) Computer Science Department, The University of Iowa, Iowa City, IA 52242, USA |
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Abstract: | A Mendelsohn triple system of order ν, MTS(ν) for short, is a pair (X, B) where X is a
ν-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct
points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered
pairs (a, b), (b, c) and (c, a). An MTS(ν) corresponds to an idempotent semisymmetric Latin square
(quasigroup) of order ν. An MTS(ν) is called frame self-orthogonal, FSOMTS for short, if its associated
semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1
n
) exists for all
n≡1 (mod 3) except n=10 and for all n≥15, n≡0 (mod 3) with possible exception that n=18. In
this paper, it is shown that (i) an FSOMTS(2
n
) exists if and only if n≡0,1 (mod 3) and n>5 with
possible exceptions n∈{9, 27, 33, 39}; (ii) an FSOMTS(3
n
) exists if and only if n≥4, with possible
exceptions that n∈{6, 14, 18, 19}.
*Research supported by NSFC 10371002
*Partially supported by National Science Foundation under Grant CCR-0098093 |
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Keywords: | Mendelsohn triple system Latin square Quasigroup Group divisible design |
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