On generalized Howell designs with block size three

In this paper, we examine a class of doubly resolvable combinatorial objects. Let $t, k, \lambda, s$ and $v$ be nonnegative integers, and let $X$ be a set of $v$ symbols. A generalized Howell design, denoted $t$-$GHD_{k}(s,v;\lambda)$, is an $s\times s$ 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 $\lambda$ 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 $\lambda=1$, 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 $n \geq 6$, 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 $n \geq 6$. 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 $\pi \in [0,5/18]$, there is a $GHD(s,v)$ whose proportion of empty cells in a row or column is arbitrarily close to $\pi$.