with the rows and columns of the array labelled by the resolution classes. In this paper, we consider generalized Howell designs, which are objects that in some sense fill in the gap between these two cases. We refer the reader to [@Handbook] for background on these objects and design theory in general.

Definition and examples {#DefnSection}
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In this paper, we examine a class of doubly resolvable designs, defined below, which generalize a number of well-known 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:

1.  \[latin\] each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of $X$);

2.  no $t$-subset of elements from $X$ appears in more than $\lambda$ cells.

In the case that $t=2$, $k=2$ and $\lambda=1$, a $2$-${\mathrm{GHD}}_2(s,v;1)$ is known as a [*Howell design*]{} $\mathrm{H}(s,v)$. In the literature, two different generalizations of Howell designs have been proposed, both of which can be incorporated into 