Three-fold Kirkman Packing Designs KPD₃({4,S～*},v)

Let X be a set of v points. A packing of X is a collection of subsets of X (called blocks) such that any pair of distinct points from X occurs together in at most A blocks in the collection. Let K be a set of positive integers. Denote by P\(K,v) a packing on v points with block sizes all in K. A packing is called resolvable if its block set admits a partition into parallel classes, each parallel class being a partition of the point set X. A Kirkman packing, denoted KP_λ(K,v), is a resolvable P_λ(K, v). A Kirkman packing design, denoted KPD_λ({w,s^*},v), is a resolvable packing of a v-set by the maximum possible number m(v) of parallel classes, each containing one block of size s and all other blocks of size w.Cerny, Horak and Walk's introduced the Kirkman packing design. Colbourn and Ling, Phillips, Walk's and Rees discussed the existence of KPD({3,s^*},v) when s ∈ {2,4}. The spectrum problem for KPD({3,4^*},v) has been almost completely solved and been used to construct perfect threshold schemes when s ≥ w by Cao and Du. Then Cao and Zhu considered the existence of KPD({3,5^*},v) when v ≡ 2 (mod 3). Since the number of parallel classes can not achieve the expired maximum, Cao and Tang considered the existence of KPD({3,4^**},v) when v ≡ 2 (mod 3). Cao and Du considered the existence of KPD({4, s^**}, v) when s ∈ {5,6}. Zhang and Du completely solved the spectrum problem for two-fold Kirkman packing designs KPD₂({3, s^*},v) when s ∈ {4,5}. In this article, we shall be restricting our attention to three-fold Kirkman packing design KPD₃({4, s^*},v) for s ∈ {5,6,7}. The following results will be proofed : there exists a KPD₃({4,5^*}, v) containing v - 3 parallel classes for every v ≡ 1 (mod 4) and v ≥ 17 ; there exists a KPD₃({4,6^*},v) containing v - 5 parallel classes for every v≡ 2 (mod 4) and v ≥ 26 ; there exists a KPD₃({4,7^*}, v) containing v - 8 parallel classes for every v ≡ 3 (mod 4) and v ≥ 51.