Frames with Block Size Four

We investigate the spectrum for frames with block size four, and discuss several applications to the construction of other combinatorial designs. Our main result is that a frame of type hu, having blocks of size four, exists if and only if u ≥ 5, h ≡ 0 mod 3 and h(u − 1) ≡ 0 mod 4, except possibly where (i) h = 9 and u ∈ {13, 17, 29, 33, 93, 113, 133, 153, 173, 193}; (ii) h ≡ 0 mod 12 and u ∈ {8, 12}, h = 36 and u ∈ {7, 18, 23, 28, 33, 38, 43, 48}, h = 24 or 120 and u ∈ {7}, h = 72 and u ∈ 2Z+ ∪ {n:n ≡ 3 mod 4 and n ≤ 527} ∪ {563}; or (iii) h ≡ 6 mod 12 and u ∈ ({17, 29, 33, 563} ∪ {n:n ≡ 3 or 11 mod 12 and n ≤ 527} ∪ {n:n ≡ 7mod 12 and n ≤ 259}), h = 18. Additionally, we give a new recursive construction for resolvable groupdivisible designs from frames: if there is a resolvable k − GDD of type gu, a k-frame of type (mg)v where u ≥ m+1, and a resolvable TD(k,mv) then there is a resolvable k − GDD of type (mg)uv . We use this to construct some new resolvable GDDs with group size three and block size four.