How to nest a perfect mendelsohn design with block size three or four

摘要

A (v, k, λ)-nested perfect Mendelsohn design, (v, k, λ)-NPMD, of form ∏_n(n~(jn), λ_n)~(i_n), 2 ≤ n ≤ k - 1, (V, B) is a (v, k, λ)-perfect Mendelsohn design with certain nested structure. In this paper some constructions for (v, 3, λ)-NPMD and (v, 4, λ)-NPMD are discussed. It is shown that the necessary conditions for the existence of such designs with block size 3 or 4 of any possible form are also sufficient, except for the non-existing designs (4, 4, λ)-NPMD of form (2~2, λ)~1, λ odd, or (8, 4, 1)-NPMD of form (2~2, 1)~1.