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abstract: |
    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 $t$-${\mathrm{GHD}}_{k}(s,v;\lambda)$, 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 ${\mathrm{GHD}}(s,v)$. 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 