## Zero capacity region of multidimensional run length constraints.(English)Zbl 0943.94004

Electron. J. Comb. 6, No. 1, Research paper R33, 16 p. (1999); printed version J. Comb. 6, 443-458 (1999).
An $$n$$-dimensional pattern of zeros and ones arranged in an $$m_1\times m_2\times\cdots\times m_n$$ hyper-rectangle is said to be $$(d,k)$$-constrained if there are at most $$k$$ consecutive zeros and between every two ones there are at least $$d$$ consecutive zeros in the binary sequence in each of the $$n$$ coordinate axis directions. The maximum number of bits of information that can be stored asymptotically per unit volume in $$n$$-dimensional space without violating the $$(d,k)$$-constraint is said to be the $$(d,k)$$-capacity. In the paper two theorems that characterize zero capacity region for finite dimensions and in the limit of large dimensions have been presented. One of them generalizes a result of the second and fourth authors [IEEE Trans. Inf.Theory 45, 1527-1540 (1999)].

### MSC:

 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory 37E99 Low-dimensional dynamical systems

Zbl 0990.59934
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