in the case $k=3$, $t=2$ and $\lambda=1$), where $e$ is then the number of empty cells in each row and column.

A Howell design with $v=s+1$, i.e. an $\mathrm{H}(s,s+1)$ is called a [*Room square*]{}. The study of Room squares goes back to the original work of Kirkman [@Kirkman] in 1850, where he presents a Room square with side length $7$, i.e. an $\mathrm{H}(7,8)$. The name of this object, however, is in reference to T. G. Room [@Room], who also constructed an $\mathrm{H}(7,8)$, and in addition showed that there is no $\mathrm{H}(3,4)$ or $\mathrm{H}(5,6)$. The existence of Room squares was settled in 1975 by Mullin and Wallis [@Mullin]; for a survey on Room squares see [@blue; @book].

\[Room squares\] There exists a Room square of side $s$ if and only if $s$ is odd and either $s=1$ or $s \geq 7$.

More generally, Stinson [@Stinson] showed existence of Howell designs with odd side $s$ and Anderson, Schellenberg and Stinson [@ASS] showed existence of Howell designs with even side $s$. We thus have the following.

\[HD\] There exists an $\mathrm{H}(s,2n)$ if and only if $n=0$ or $n \leq s \leq 2n-1$, $(s,2n)\not\in \{(2,4), (3,4), (5,6), (5,8)\}$.

A $2$-${\mathrm{GHD}}_k(s,v;\lambda)$ with $s=\frac{\lambda(v-1)}{k-1}$ is equivalent 