Group divisible covering designs with block size four

Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k-GDCD of type g^u is a covering number denoted by C(k, g^u). When k = 3, the values of C(3, g^u) have been determined completely for all possible pairs (g, u). When k = 4, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4-IGDDs are constructed, thereby completing the existence problem for 4-IGDDs of type (g, h)^u. Then, additional families of optimal 4-GDCDs are constructed. Consequently the cases for (g, u) whose status remains undetermined arise when g ≡ 11, 14, 17, 23 and 24, when u ≡ 5 and 6, and in several small families for which one of g and u is fixed.