FRAMES WITH BLOCK SIZE-4

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 greater-than-or-equal-to 5, h = 0 mod 3 and h(u - 1) = 0 mod 4, except possibly where (i) h = 9 and u is-an-element-of {13,17.29,33,93,113,133,153,173,193}; (ii) h = 0 mod 12 and u is-an-element-of {8,12}, h = 36 and u is-an-element-of {7,18,23,28,33,38,43,48}, h = 24 or 120 and u is-an-element-of {7}, h = 72 and u is-an-element-of 2Z+ or {n : n = 3 mod 4 and n less-than-or-equal-to 527} or {563}; or (iii) h = 6 mod 12 and u is-an-element-of ({17,29,33,563} or {n : n = 3 or 11 mod 12 and n less-than-or-equal-to 527} or {n : n = 7 mod 12 and n less-than-or-equal-to 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 greater-than-or-equal-to 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.