Minimum embedding of a P4P4-design into a balanced incomplete block design of index λλ

Let HH be a subgraph of GG. An HH-design (V,C)(V,C) of order vv and index μμ is embedded   into a GG-design (X,B)(X,B) of order v+wv+w and index λλ if μ≤λμ≤λ, V⊆XV⊆X and there is an injective mapping f:C→B such that BB is subgraph of f(B) for every B∈CB∈C.For every pair of positive integers vv, λλ, (except when λ=3λ=3 and v=30,34,42,46,54,58,66v=30,34,42,46,54,58,66 or λ=5λ=5 and v=19v=19) we determine the minimum value of ww such that there exists a balanced incomplete block design of order v+wv+w, index λλ and block-size 4 which embeds a P4P4-design of order vv and index μ=1μ=1 (P4P4 denotes the path of length 3).