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

TL;DR

This paper extends the concept of entropy-based capacity bounds from 1-D to 2-D constraints using a concave programming approach, providing improved upper bounds for specific 2-D constraints.

Contribution

It introduces a novel concave programming method to compute upper bounds on 2-D constraint capacities, generalizing 1-D entropy methods.

Findings

Improved upper bounds for 2-D 'no independent bits' constraint.

Enhanced bounds for certain 2-D RLL constraints.

Method is numerically efficient and adaptable.

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.