Some New Results On Frames

Let K be a set of positive integers. A group divisible design (K,λ)-GDD is a triple (χ,(?),β) which X is a finite set of points, (?) is a partition ofχinto subsets called groups,βis a collection of subsets ofχ(called blocks) with sizes from K, such that every pair of points from distinct groups occurs in exactlyλblocks, and no pair of points belonging to a group occurs in any block.A (K,λ)-frame is a (K,λ)-GDD (χ, (?),β) whichβcan be partitioned into holey parallel classes, each holey parallel class being a partition ofχ\ G_j for some G_j∈(?). A (3,λ)-frame is also called a Kirkman frame with indexλ. A frame is called simple if all its blocks are distinct.Letχbe a set of tmn points, which is partitioned into m-subsets X_ij, 0≤i≤n -1,0≤j≤t-1.A holey group divisible design, denoted by (k,λ)-HGDD of type (n, m^t), is an ordered pair (χ,β) whichβbe a collection of subsets ofχ(called blocks), such that| B|= k for every block B∈β, every pair of points x∈X_i1j1 and y∈X_i2j2 is contained in exactly A blocks, where i₁≠i₂ and j₁≠j₂, and the pair of points x and y is not contained in any block if i₁ = i₂ or j₁ = j₂. When m = 1, a (k,λ)-HGDD is called a modified group divisible design (MGDD). A modified (k,λ)-frame of type g^u is a (k,λ)-MGDD whose blocks can be partitioned into holey parallel classes, each of which is respect to some group.A double frame (k,λ)-DF of type (n, m^t) is a (k,λ)-HGDD of type (n, m^t) (χ,β) withχ= {X_ij :0≤i≤n-1,0≤j≤t-1.} andβcan be partitioned into holey parallel classes each of which partitions∪_{p≠i,q≠j}X_pq for some i and j, 0≤i≤n -1,0≤j≤t-1.Frames are powerful for the construction of other types of combinatorial designsin design theory. In this thesis, we mainly investigate simple frames and double frames. We shall prove that the necessary conditions for simple Kirkman frames with index 4 are also sufficient and give a complete solution for the existenceof (4,λ)-HGDDs and (3,λ)-double frames. Further, we shall obtain some new results on (4, 1)-modified frames and resolvable modified group divisible designs.