New Bounds on the Capacity of Multidimensional Run-Length Constraints

Abstract

Abstract-We examine the well-known problem of determining the capacity of multidimensional run-length-limited constrained systems. By recasting the problem, which is essentially a combinatorial counting problem, into a probabilistic setting, we are able to derive new lower and upper bounds on the capacity of (0; k)-RLL systems. These bounds are better than all previously-known analytical bounds for k 2, and are tight asymptotically. Thus, we settle the open question: what is the rate at which the capacity of (0; k)-RLL systems converges to 1 as k ! 1? We also provide the first nontrivial upper bound on the capacity of general (d; k)-RLL systems.

Semiconstrained systems

“…We note that taking c = 0 in Theorem 28 gives 1 − C k,0 (1)), which coincides with the capacity's rate of convergence for the fully-constrained system [24].…”

“…Following the same logic as [24], we replace the combinatorial counting problem with a probability-bounding problem. Assume p n denotes the probability that a random string from Σ n , which is chosen with uniform distribution, is in B n (F , P).…”

“…We can extend the method of monotone families that was used in [24] to obtain such a bound. However, the result that we describe next, which is based on the theory of large deviations, outperforms the monotone-families approach.…”

“…Thus, the expression may be further simplified by plugging in lower bounds on the one-dimensional capacity from Theorem 29 or Theorem 30. We follow the steps presented in [24], and therefore, only sketch the proof. As a final comment we note that the ratio between the bounds of Theorem 28 and Theorem 30 is at most ≈ 1.5.…”

“…This rate of convergence is known when the system is fully constrained [24]. Additionally, the capacity is known only in the one-dimensional case, whereas the general bounds may be extended to the multidimensional case as well.…”