Improved Bit-Stuffing Bounds on Two-Dimensional Constraints

Abstract-A stationary model is presented for the twodimensional (2-D) no isolated bits (n.i.b.) constraint over an extended alphabet dened by the elements within 1 by 2 blocks. This block-wise model is based on a set of sufcient conditions for a Pickard random eld (PRF) over an m−ary alphabet. Iterative techniques are applied as part of determining the model parameters. Given two Markov chains describing a boundary, an algorithm is presented which determines whether a certain PRF consistent with the boundary exists. Iterative scaling is used as part of the algorithm, which also determines the conditional probabilities yielding the maximum entropy for the given boundary description if a solution exists. Optimizing over the parameters for a class of boundaries with certain symmetry properties, an entropy of 0.9156 is achieved for the n.i.b. constraint, providing a lower bound. An algorithm for iterative search for a PRF solution starting from a set of conditional probabilities is also presented.

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“…constraint. It is slightly higher than the lower bound of 0.9127 presented as a lower bound for bit-stufng in [7].…”

Section: ) Results Using Iterative Scalingcontrasting

confidence: 54%

“…constraint is not known, but in [7] it was estimated to be 0.9238 and lower bounds on the entropy obtained by bit-stufng were given. For a rectangle to represent the forbidden words, the n.i.b.…”

Section: A Constrained Eldsmentioning

confidence: 99%

A Model for the Two-Dimensional No Isolated Bits Constraint

Forchhammer

Laursen

2006

2006 IEEE International Symposium on Information Theory

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“…Using the same method in the 3-D case, it was shown in [18] that Several constructive methods were suggested which, by devising an appropriate encoder and analyzing its rate, yield lower bounds on the capacity of the constrained system. These works 0018-9448/$26.00 © 2011 IEEE include [22], [25], [26] as well as the bit-stuffing method described in [7], [27]. A few recursive constructions were also given by Etzion in [5].…”

Section: Introductionmentioning

confidence: 99%

New Bounds on the Capacity of Multidimensional Run-Length Constraints

Schwartz

Vardy

2011

IEEE Trans. Inform. Theory

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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.

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“…There exists a configuration (u i,j ) (i,j)∈A over the vertex set V such that (a) for each (i, j) ∈ A we have w i,j = L(u i,j ); (b) each row in (u i,j ) is a path in G row ; (c) each column in (u i,j ) is a path in G col . Examples of 2-D constraints include the square constraint [2], 2-D runlength-limited (RLL) constraints [3], 2-D symmetric runlength-limited (SRLL) constraints [4], and the "no isolated bits" constraint [5].…”

Section: Introductionmentioning

confidence: 99%

Convex Programming Upper Bounds on the Capacity of 2-D Constraints

Tal

Roth

2011

IEEE Trans. Inform. Theory

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Abstract-The capacity of 1-D constraints is given by the entropy of a corresponding stationary maxentropic Markov chain. Namely, the entropy is maximized over a set of probability distributions, which is defined by some linear requirements. In this paper, certain aspects of this characterization are extended to 2-D constraints. The result is a method for calculating an upper bound on the capacity of 2-D constraints.The key steps are: The maxentropic stationary probability distribution on square configurations is considered. A set of linear equalities and inequalities is derived from this stationarity. The result is a concave program, which can be easily solved numerically. Our method improves upon previous upper bounds for the capacity of the 2-D "no independent bits" constraint, as well as certain 2-D RLL constraints.

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Improved Bit-Stuffing Bounds on Two-Dimensional Constraints

Cited by 52 publications

References 24 publications

A Model for the Two-Dimensional No Isolated Bits Constraint

A Model for the Two-Dimensional No Isolated Bits Constraint

New Bounds on the Capacity of Multidimensional Run-Length Constraints

Convex Programming Upper Bounds on the Capacity of 2-D Constraints

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