简单自反MTS(υ,λ)的存在谱

THE SPECTRUM OF SIMPLE SELF-CONVERSE MTS (υ,λ)

一个Mendelsohn三元系MTS(υ,λ)=(X,B)被称作是自反的,如果它与它的逆(X,B-1)是同构的,其中B-1=｛〈z,y,x〉;〈x,y,z〉∈B.在[2]中已给出了简单自反MTS((υ,1)的存在谱,即υ≡0,1(mod3),υ3且υ≠6.本文讨论一般λ的情况,并得到简单自反MTS(υ,λ)的存在谱是λυ(υ-1)≡0(mod3);λυ-2,υ3且(υ,λ)≠(6,1);(6,3).

Let X be a υ-set, v 3. A cyclic triple from X is a collection of three ordered pairs (x, y) , (y, z) and (z, x), where x, y, z are distinct elements of X. It is denoted by (x, y, z)(or (y, z, x) or (z, x, y)). A Mendelsolsn triple system on X is a pair (X, B) where B is a collection of some cyclic triples from X such that each ordered pair of X appears in λ blocks,denoted by MTS (υ,λ). A Mendelsohn triple system MTS(υ,λ) = (X, B) is called selfconverse if there is a permutation f on X which maps B onto B-∞= {B-∞=~ (,, §); B =(§, , ) ∈B}. We denote a self-converse MTS(υ,λ) by SCMTS(υ,λ)=(X, B, {). A design called simple if it has no repeated blocks. The problem of the existence spectrum of SCMTS was posed by C.J. Colbouru and A. Rosa in their survey. The case of λA=1 was given, i.e.υ=0,1 (mod 3), υ 3 and υ≠6. In this paper we discuss the case of general λ and give a better result-the spectrum of simple SCMTS(υ,λ), i.e. λυ(υ-1)≡0 (mod 3),λυ-2, υ 3 and (υ,λ)≡(6, 1), (6,3).