Embedding handcuffed designs with block size

Let (W, C) be an m-cycle system of order n and let Ω ⊂ W , |Ω| = v < n. We say that a handcuffed design (Ω,P) of order v and block size s (2 ≤ s ≤ m−1) is contained in (W, C) if for every p ∈ P there is an m-cycle c = (a1, a2, . . . , am) ∈ C such that: (1) p = [ak, ak+1, . . . , ak+s−1] for some k ∈ {1, 2, . . . ,m} (i.e. the (s− 1)-path p occurs in the m-cycle c); and (2) ak−1, ak+s ∈ Ω. Note that in (1) and (2) all the indices are reduced to the range {1, . . . ,m} (mod m). For each n ≡ 1 (mod 8) and for each s ∈ {2, 3} we determine all the integers v such that there is a 4-cycle system of order n containing a handcuffed design of order v and block size s.