On lattice H design with block size four

A group divisible t-design (X, G, B) is called a lattice group divisible t-design, if for any block B ∈ B and any column G′j, either |B ∩ G′j| < t or B ⊆ G′j. When t = 3 and block size is four, we denote a lattice group divisible 3-design of type gⁿ by LH(n, g, 4, 3). The necessary conditions on the existence of an LH(n, g, 4, 3) are n ≥ 4 and n ≡ 2, 4(mod 6). In this paper, we show that the necessary conditions are also sufficient. An LH(n, g, 4, 3) is said to be resolvable, and denoted by RLH(n, g, 4, 3), if its block set can be partitioned into parts, such that each point of the design occurs in exactly one block in each part. We also show that an RLH(n, g, 4, 3) exists if and only if n ≡ 4, 8(mod 12).