The 1-vertex transfer matrix and accurate estimation of channel capacity

TL;DR

This paper introduces a 1-vertex transfer matrix approach that efficiently estimates the capacity of multi-dimensional codes, significantly reducing computational resources while improving bounds for the 2D (0,1) run length limited channel.

Contribution

The paper presents a novel 1-vertex transfer matrix method for estimating channel capacity, offering computational efficiency and improved bounds over traditional methods.

Findings

Efficient estimation of 2D channel entropy to 15 digits

Reduced storage and computation compared to standard transfer matrices

Improved bounds for the (0,1) run length limited channel in 2 and 3 dimensions

Abstract

The notion of a 1-vertex transfer matrix for multi-dimensional codes is introduced. It is shown that the capacity of such codes, or the topological entropy, can be expressed as the limit of the logarithm of spectral radii of 1-vertex transfer matrices. Storage and computations using the 1-vertex transfer matrix are much smaller than storage and computations needed for the standard transfer matrix. The method is applied to estimate the first 15 digits of the entropy of the 2-dimensional (0,1) run length limited channel. In order to compare the computational cost of the new method with the standard transfer matrix and have rigorous bounds to compare the estimates with a large scale computation of eigenvalues for the (0,1) run length limited channel in 2 and 3 dimensions have been carried out. This in turn leads to improvements on the best previous lower and upper bounds for that channel.