On generalized Howell designs with block size three

In this paper, we examine a class of doubly resolvable combinatorial objects. Let t, k, λ, s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHDk(s, v;λ), is an s × s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than λ cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes. In this paper, we concentrate on the case that t = 2, k = 3 and λ = 1, and write GHD(s, v). In this case, the number of empty cells in each row and column falls between 0 and (s−1)/3. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least (s − 2)/3 empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a GHD(n+1, 3n) if and only if n ≥ 6, except possibly for n = 6. In the case of two empty cells, we show that there exists a GHD(n+ 2, 3n) if and only if n ≥ 6. Noting that the proportion of cells in a given row or column of a GHD(s, v) which are empty falls in the interval [0, 1/3), we prove that for any π ∈ [0, 5/18], there is a GHD(s, v) whose proportion of empty cells in a row or column is arbitrarily close to π.