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.
Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.
Abstract A t-hyperwhesl (t 〉 3) of length l (or Wz(t) for brevity) is a t-uniform hypergraph (V, E), where t E= {e1,e2,...,el} and vl,v2,...,vt are distinct vertices of V = Ui=1 ei such that for i= 1,...,1, vi,vi+1 ∈ei and ei ∩ ej = P, j ∈ {i - 1, i,i + 1}, where the operation on the subscripts is modulo 1 and P is a vertex of V which is different from vi, 1 〈 i 〈 l. In this paper, the minimum covering problem of MCλ(3, W(3),v) is investigated. Direct and recursive constructions on MCλ(3, W(3),v) are presented. The covering number cλ(3, W4(3), v) is finally determined for any positive integers v 〉 5 and A.
In this paper, several recursive constructions for directed difference family and perfect directed 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 difference family in Zgv are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type gv are obtained.