Randomized Latin-square design

Fix $t\ge 1$ and put $R_t=\{0,1,\dots ,t-1\}$, with addition interpreted modulo $t$. A Latin-square allocation of order $t$ is a function $L:R_t\times R_t\to R_t$ such that for each fixed row the map from columns to treatments is a bijection, and for each fixed column the map from rows to treatments is a bijection. Thus each treatment occurs exactly once in every row and every column. Let ${\mathcal{L}}_t$ be the finite set of these allocations and assign the uniform law on ${\mathcal{L}}_t$. This law is well-defined: $L_0(r,c)=r+c(\mathrm{m}\mathrm{o}\mathrm{d}t)$ is a Latin square, since adding a fixed residue modulo $t$ permutes the residues; hence ${\mathcal{L}}_t$ is nonempty. This specifies one exact restricted randomization law. It randomizes over treatment-by-row-by-column assignments and, for $t\ge 2$, is distinct from the product of independent within-block permutations in an RCBD (Randomized complete-block design).