Zero capacity region of multidimensional run length constraints

Run length constraints derive from digital storage applications. For nonnegative integers d and k, a binary sequence is (d,k)-constrained if there are at most k consecutive zeros and between every two ones there are at least d consecutive zeros. We present two main results that characterize the zero capacity region for finite dimensions and in the limit of large dimensions. The first result generalizes the zero capacity characterization of Kato and Zeger (see IEEE Trans. Inform. Theory. vol.45, no.4, p.1527-40, 1999) to all dimensions greater than one, which turns out to be exactly the same as in dimension 2. The second result gives a necessary and sufficient condition on d and k, such that the capacity approaches zero in the limit as the dimension n grows to infinity