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 gum1, where g ≡ 2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type gum1 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 = 2tqs, q  = 19, 23, 25, 29, 31, s , t =1,2,… , as well as for g = 2tqs, 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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