Frames with Block Size Four

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Abstract
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+ U ﹛n : n ≡ 3 mod4 and n ≤527﹜ U ﹛563﹜; or (iii)h ≡ 6mod l2 and u ∈ (﹛17,29,33,563﹜ U ﹛n : n ≡ 3 or 11 mod 12 and n ≤ 527﹜ U ﹛n : n ≡ 7 mod 12 and n ≤ 259﹜), h = 18. Additionally, we give a new recursive construction for resolvable group-divisible 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.

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