The Existence Of Simple Strictly Cyclic Triple Systems

Let v,k,λ be positive integers.β is a collection of k-subsets(called blocks)from a v-set V,such that any given pair of elements in V lies in exactly λ blocks,then(V,β)is called balanced incomplete block design,denoted by B(k,λ；v).A B(k,λ；v)is simple if it contains no repeated blocks.A B(k,λ；v)is cyclic if it has an automorphism consisting of a single cycle of lengthv.If a cyclic B(k,λ；v)doesn’t contain short orbits,it is called strictly cyclic.A triple system is a B(k,λ；v)with k=3,denoted by TS(v,λ).We use the notation SSCTS(v,λ)denote a simple strictly cyclic TS(v,λ).In the paper,we discuss the existence problem of SSCTS(v,λ)by directly constructing with difference triples,and obtain the following results: vis a Odd:(1)An SSCTS(V,λ)exists for any v≡1(mod6),v≥13,λ=6,7,8,9.(2)An SSCTS(v,λ)exists for any v≡3,5(mod6),v≥11,A=6,9. vis a even:(1)When v=2p,p is a prime: For any p≡1(mod6),an SSCTS(2p,λ)exists for any λ satisfying the necessary con-ditions；Furthermore,when p≥31,g-1≠g2(g+1),a DSSCTS(2p；124(2p-50)1) exists,where g is the generater of the unit group U2p of Z2p. For any p≡5(mod6),an SSCTS(2p,12t)exists for1≤t≤p5/6,and an SSCTS(2p,12t+8)exists for0≤f≤p-5/6.(2)When v=2p.,prime p≡1(mod6),p≥19,a DSSCTS(2p2；122(12p)1(2p212p-26)1)exists. Furthermore,an SSCTS(2p2,12t)exists for t∈A∪B,where A={1,2,p,p+1,p+2),B={2p2-2-12a:a∈A).