Roman and Vatican Crossover Designs

Latin squares with a balance property among adjacent pairs of symbols---being "Roman" or "row-complete"---have long been used as uniform crossover designs with the number of treatments, periods and subjects all equal. This has been generalized in two ways: to crossover designs with more subjects and to balance properties at greater distances. We consider both of these simultaneously, introducing and constructing {\em Vatican designs}: these have $\ell t$ subjects, $t$ periods and treatments, and, for each $d$ in the range $1 \leq d < t$, the number of times that any subject receives treatment $j$ exactly $d$ periods after receiving treatment $i$ is at most $\ell$. Results include showing the existence of Vatican designs when $t+1$ is prime (for any $\ell$), when $5 \leq t \leq 14$ and $\ell >1$, and when $t \in \{ 3,15 \}$ and $\ell$ is even.