  De Bruijn tori, also called perfect maps, are two-dimensional periodic arrays
of letters drawn from a given finite alphabet, such that each possible pattern
of a given shape $(m,n)$ appears exactly once within one period of the torus.
It is still unknown if de Bruijn tori of some certain size exist, like e.g.
square shaped de Bruijn Tori with odd $m=n\in\{3,5,7\}$ and an even alphabet
size $k$. However, in certain applications like positional coding, sub-perfect
maps are sufficient, i.e. one does not need every possible $(m,n)$-pattern to
appear, as long as a sufficient large number of such patterns is captured and
every pattern occurs at most once. We show, that given any $m=n$ and a square
alphabet size $k^2$, one can efficiently construct a sub-perfect map which is
almost perfect, i.e. of almost maximal size. We do this by introducing de
Bruijn rings, i.e. sub-perfect maps of minimal height, and providing an
efficient construction method for them. We extend our results to non-square
torus shapes and arbitrary non-prime alphabet sizes.
