  This paper studies two families of constraints for two-dimensional and
multidimensional arrays. The first family requires that a multidimensional
array will not contain a cube of zeros of some fixed size and the second
constraint imposes that there will not be two identical cubes of a given size
in the array. These constraints are natural extensions of their one-dimensional
counterpart that have been rigorously studied recently. For both of these
constraint we present conditions of the size of the cube for which the
asymptotic rate of the set of valid arrays approaches 1 as well as conditions
for the redundancy to be at most a single symbol. For the first family we
present an efficient encoding algorithm that uses a single symbol to encode
arbitrary information into a valid array and for the second family we present a
similar encoder for the two-dimensional case. The results in the paper are also
extended to similar constraints where the sub-array is not necessarily a cube,
but a box of arbitrary dimensions and only its volume is bounded.
