objects, see [@Handbook].

We say that $(X, \cB)$ is a [*pairwise balanced design*]{} PBD$(v, K,\lambda)$ if $X$ is a set of $v$ elements and $\cB$ is a collection of subsets of $X$, called [*blocks*]{}, which between them contain every pair of elements of $X$ exactly $\lambda$ times. If $K=\{k\}$, then a $\mathrm{PBD}(v,\{k\},\lambda)$ is referred to as a [*balanced incomplete block design*]{}, ${\mathrm{BIBD}}(v,k,\lambda)$. A collection of blocks that between them contain each point of $X$ exactly once is called a [*resolution class*]{} of $X$. If $\cB$ can be partitioned into resolution classes we say that the design is [*resolvable*]{}, and refer to the partition as a [*resolution*]{}.

It is possible that a design may admit two resolutions, $\cR$ and $\cS$. If $|R_i\cap S_j| \leq 1$, for every resolution class $R_i\in \cR$ and $S_j\in \cS$, we say that these resolutions are [*orthogonal*]{}. A design admitting a pair of orthogonal resolutions is called [*doubly resolvable*]{}.

In the definition of a PBD or BIBD, if we relax the constraint that every pair appears exactly once to that every pair appears at most once, then we have a [*packing*]{} design. Thus, a $2$-${\mathrm{GHD}}_k(s,v;\lambda)$ may be viewed as a doubly resolvable packing design.

A [*group-divisible design*]{}, 