In this paper, it is shown that the necessary conditions for the existence of a ( gv, {g, 3 α }, 3, λ)-DF in Z gv for α∈ {0, 1, 2} are also sufficient with two exceptions of (v, g, λ, α) = (9, 1, 1, 1), (9, 1, 2, 2). Finally, the existence spectrum of a cyclic (3, λ)-GDD of type g v is determined.
A t-hyperwheel(t≥3) of length l(or W(t) l for brevity) is a t-uniform hypergraph(V,E),where E = {e 1,e 2,...,e l } and v 1,v 2,...,v l are distinct vertices of V = ∪ei i=1 l such that for i = 1,...,l,v i,v i+1 ∈ei and e i∩ej = P,j ∈/{i 1,i,i + 1},where the operation on the subscripts is modulo l and P is a vertex of V which is different from v i,1 ≤ i ≤ l.In this paper,the minimum covering problem of M Cλ(3,W(3)4,v) is investigated.Direct and recursive constructions on M C λ(3,W(3) 4,v) are presented.The covering number cλ(3,W(3)4,v) is finally determined for any positive integers v ≥ 5 and λ.
In this paper, several recursive constructions for directed difference family and perfectdirected difference family are presented by means of difference matrix and incomplete difference matrix.Finally the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)-directed differencefamily in Z_(gv) are established. As a consequence, the necessary and sufficient conditions for the existenceof a cyclic directed group divisible design with block size three and type g^v are obtained.
