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

Forbes, A. D. (2019). Group divisible designs with block size four and type gum1—II. Journal of Combinatorial Designs, 27(5) pp. 311–349.

DOI: https://doi.org/10.1002/jcd.21650

Abstract

We show that the necessary conditions for the existence of group divisible designs with block size four (4‐GDDs) of type gum1 are sufficient for g ≡ 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 ≡ 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 ≡ 1 (mod 4) or d is a prime not greater than 43. Hence there are no exceptions when g ≡ 3 (mod 12). Consequently, we are able to extend the known spectrum for g ≡ 1 and 5 (mod 6). Also, we complete the spectrum for 4‐GDDs of type (3α)4(6α)1(3b)1.

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