Frames with Block Size Four

Source abstract

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 h u , 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 g u , 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.