Group Divisible Designs, GBRSDS And Generalized Weighing Matrices

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p – 1 we show the following exist: (i) GDD(2(p2 +p+ 1), 2(p2 +p+ 1), rp2, 2p2, λ1 = p2λ, λ2 = (p2 —p)r, m = p2 + p+ 1, n = 2), r = 1,2; (ii) GDD(q(p+ 1), q(p+ 1), p(q – 1), p(q –1), λ1 = (q – 1)(q – 2), λ2 = (p– 1)(q – 1)2/q, m = q, n = p+1); (iii) PBD(21, 10; K), K = {3, 6, 7} and PBD(78, 38; K), K = {6, 9, 45}; (iv) GW(vk, k2; EA(k)) whenever a (v, k, λ)-difference set exists and k is a prime power; (v) PBIBD(vk2, vk2, k2, k2; λ1 = 0, λ2 = λ, λ3 = k) whenever a (v, k, λ)-difference set exists and k is a prime power; (vi) we give a GW(21; 9; Z3). The GDD obtained are not found in W.H. Clatworthy, “Tables of Two-Associate-Class, Partially Balanced Designs”, NBS, US Department of Commerce, 1971.