the definition above: these are due to Deza and Vanstone [@DezaVanstone] (which corresponds to the case that $t=2$) and to Rosa [@Rosa] (which corresponds to the case that $t=k$). The objects in question have appeared under several names in the literature. The term [*generalized Howell design*]{} appears in both [@DezaVanstone] and [@Rosa] in reference to the particular cases studied in these papers, and also in papers such as [@WangDu]. The term [*generalized Kirkman square*]{} or GKS has more recently been introduced to the literature (see [@DuAbelWang; @Etzion]); in particular, in these papers, a $\mathrm{GKS}_k(v;1,\lambda;s)$ is defined to be what we have defined as a $2$-${\mathrm{GHD}}_k(s,v;\lambda)$. In this paper we will concentrate on the case when $t=2$, $k=3$ and $\lambda=1$, in which case we omit these parameters in the notation and simply write ${\mathrm{GHD}}(s,v)$, or ${\mathrm{GHD}}_k(s,v)$ for more general $k$.

Two obvious necessary conditions for the existence of a non-trivial $2$-${\mathrm{GHD}}_k(s,v;\lambda)$ are that $v\equiv 0 \pmod k$ and that $\frac{v}{k}\leq s \leq \frac{\lambda(v-1)}{k-1}$. In particular, when $k=3$, $t=2$ and $\lambda=1$, we have that $\frac{v}{3}\leq s \leq \frac{v-1}{2}$. Since a $t$-${\mathrm{GHD}}_k(s,v;\lambda)$ contains exactly $n=\frac{v}{k}$ non-empty cells in each row and column, it can be helpful to write $t$-${\mathrm{GHD}}_k(n+e, kn;\lambda)$ (or ${\mathrm{GHD}}(n+e, 3n)$ 