Group divisible designs with block size four and type gum1-II: FORBES

We show that the necessary conditions for the existence of group divisible designs with block size four (4-GDDs) of type g(u)m(1) are sufficient for g equivalent to 0 (mod h), h = 39, 51, 57, 69, 87, 93, 111, 123 and 129, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for g equivalent to 3 (mod 6), the possible exceptions occur only when u = 8, and there are no exceptions at all if g/3 has a divisor d > 1 such that d equivalent to 1 (mod 4) or d is a prime not greater than 43. Hence there are no exceptions when g equivalent to 3 (mod 12). Consequently, we are able to extend the known spectrum for g equivalent to 1 and 5 (mod 6). Also, we complete the spectrum for 4-GDDs of type (3a)(4)(6a)(1)(3b)(1).