Directed Kirkman Packing Designs DKPD({3,5～*},v)

Let X be a set of v points. A packing(directed packing) of X is a collection ofsubsets(ordered subsets) of X (called blocks) such that any pair(ordered pair) of distinctpoints from X occurs together in at most one block in the collection. A packing(directedpacking) is called resolvable if its block set admits a partition into parallel classes, eachparallel class being a partition of the point set X. A Kirkman packing design, denotedKPD({w,s^*},v), is a resolvable packing of a v-set by the maximum possible numberm(v) of parallel classes, each containing one block of size s and all other blocks of sizew. A directed Kirkman packing design, denoted DKPD({w,s^*},v), is a resolvabledirected 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.(?)er(?)y, Horák and Wallis introduced the Kirkman packing design. Colbourn and Ling,Phillips, Wallis and Rees discussed the existence of KPD({3,s^*},v) when s∈{2,4}. Thespectrum problem for KPD({3, 4^*}, v) has been almost completely solved and been usedto construct perfect threshold schemes when s≥w by Cao and Du. Then Cao and Zhuconsidered the existence of KPD({3,5^*},v) when v≡2 (mod 3). Since the number ofparallel classes can not achieve the expired maximum, Cao and Tang considered theexistence of KPD({3,4^**},v) when v≡2 (mod 3). Cao and Du considered the existen-ce of KPD({4,s^*},v) when s∈{5,6}. Zhang and Du completely solved the spectrumproblem for the directed Kirkman packing design DKPD({3,s^*},v) when s∈{2,4}. Inthis article, we shall be restricting our attention to the directed Kirkman packing designDKPD({3,5^*},v). The following result will be proved: there exists a DKPD({3,5^*},v)containing v-6 parallel classes for every v≡2 (rood 3) and v≥26.