Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let Kq(n, w, t, d) be the minimum size of a code over Zq of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine Kq(n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2m + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.