Group divisible designs with block size four and type gum1—III

Forbes, A. D. (2019). Group divisible designs with block size four and type gum1—III. Journal of Combinatorial Designs, 27(11) pp. 643–672.

DOI: https://doi.org/10.1002/jcd.21668

Abstract

We deal with group divisible designs (GDDs) that have block size four and group type g^um¹, where g ≡ 2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type g^um¹ are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for g = 2^tq^s, q  = 19, 23, 25, 29, 31, s , t =1,2,… , as well as for g = 2^tq^s, q = 2, 5, 7, 11, 13, 17, ????=1,2,… , with possible exceptions 56⁹m¹, 80⁹m¹ and 112⁹m¹ for a few large values of m.

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