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

Abstract

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.

A Bit-Stuffing Algorithm for Crosstalk Avoidance in High Speed Switching

“…There are recent efforts in obtaining bounds by linear programming and convex programming for the bit-stuffing algorithms on various applications (see e.g., [21], [22]). It would be of interest to apply their approaches to obtain tight bounds for our setting.…”

On the limits of communication over optical on-off keying channels with crosstalk

“…In general, due to the high computational complexity, few studies have been conducted to obtain tight bounds on the capacity of 2-D channels with memory, except for the capacity of some special 2-D constraints. Specifically, the lower bounds on the capacity of some 2-D constraints were presented and analyzed in [15]- [17], based on either bit-stuffing encoders or tilling encoders; and the upper bounds were provided by Forchhammer and Justesen [18], Tal and Roth [19].…”

“…We then extend the idea for the 1-D case to compute an upper bound on the capacity for the 2-D case. In [19], it was demonstrated that there exists a stationary and symmetric input distribution that achieves the capacity of some special 2-D constraints. In fact, we can prove that this conclusion also holds for optical on-off-keying channels with 2-D crosstalk.…”

Patterned cells for phase change memories

“…In this figure, the vertex i is marked as a gray square; D i is indicated by the black vertices that the vertex i depends on; the stationary constraint is applied to the region T that includes all the vertices plotted. Based on these schemes, we get the lower bounds for the capacities, which are given in the second column in Table I. 3) Upper Bound based on Convex Programming: In [6], convex programming was used as a method for calculating an upper bound on the capacity of 2-D constraints. The idea is based on the observations that there exists an optimal distribution θ * such that θ * is stationary and symmetric when the array is sufficiently large.…”

“…For a rectangular array, if two sets of vertices A and B are reflection symmetric about a horizontal/vertical line or a 45-degree line, then they have the same state (configuration) distribution. Note that the reflection symmetry about a 45-degree line is also called transposition invariance in [6]. For a triangular array, there are more symmetries: if two sets of vertices A and B are reflection symmetric about a horizontal/vertical line or a 30/60-degree line, then they have the same state (configuration) distribution.…”