Building upon the work of Wei and Ge (Designs, Codes, and Cryptography 74, 2015), we extend the range of positive integer parameters g, u, and m for which group divisible designs with block size 4 and type gum1 are known to exist. In particular, we show that the necessary conditions for the existence of these designs when m>0 and m≠g are sufficient in the following cases: g≡0( mod 6), g=2 with one exception, 2651, g=10, and g=11.

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 ≡ 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 ( 3 a ) 4 ( 6 a ) 1 ( 3 b ) 1.

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.

Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4$\,$075 designs with chromatic index 11 and two with chromatic index 12; all are 3-resolvable; and there are exactly two 3-existentially closed STS(19).

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