to a [*doubly resolvable balanced incomplete block design*]{} ${\mathrm{BIBD}}(v,k,\lambda)$. Doubly resolvable designs and related objects have been studied, for example, in [@OCD; @Curran; @Vanstone; @Fuji-Hara; @Vanstone; @Fuji-Hara; @Vanstone; @TD; @Vanstone; @AG; @Lamken; @09; @Vanstone; @80; @Vanstone; @82]. In particular, Fuji-Hara and Vanstone investigated orthogonal resolutions in affine geometries, showing the existence of a doubly resolvable ${\mathrm{BIBD}}(q^n,q,1)$ for prime powers $q$ and integers $n \geq 3$. Asymptotic existence of doubly resolvable ${\mathrm{BIBD}}(v,k,1)$ was shown by Rosa and Vanstone [@Rosa; @Vanstone] for $k=3$ and by Lamken [@Lamken; @09] for general $k$.

For $t=2$, $k=3$ and $\lambda=1$, a ${\mathrm{GHD}}(s,v)$ with $s = \lfloor\frac{v-1}{2}\rfloor$ corresponds to a [*Kirkman square*]{}, ${\mathrm{KS}}(v)$, (i.e. a doubly resolvable Kirkman triple system of order $v$) when $v \equiv 3 \pmod 6$ and a [*doubly resolvable nearly Kirkman triple system*]{}, $\mathrm{DRNKTS}(v)$, when $v\equiv 0 \pmod 6$. A Kirkman square of order $3$ is trivial to construct. Mathon and Vanstone [@Mathon; @Vanstone] showed the non-existence of a ${\mathrm{KS}}(9)$ or ${\mathrm{KS}}(15)$, while the non-existence of a $\mathrm{DRNKTS}(6)$ and $\mathrm{DRNKTS}(12)$ follows from Kotzig and Rosa [@KotzigRosa74]. For many years, the smallest known example of a ${\mathrm{GHD}}(s,v)$ with $s = \lfloor\frac{v-1}{2}\rfloor$ (other than the trivial case of $s=1$, $v=3$) was a $\mathrm{DRNKTS}(24)$, found by 