Finite size effects and error-free communication in Gaussian channels

TL;DR

This paper investigates a Gallager-type error-correcting code for Gaussian channels, demonstrating that finite size effects and low connectivity can nearly reach Shannon's theoretical limits in error-free communication.

Contribution

It introduces a novel code construction using complex matrices and analyzes finite size effects, showing near-optimal performance close to Shannon bounds even with low connectivity.

Findings

Finite size effects are quantitatively estimated.

Critical noise levels nearly saturate Shannon bounds.

Low connectivity codes achieve near-optimal performance.

Abstract

The efficacy of a specially constructed Gallager-type error-correcting code to communication in a Gaussian channel is being examined. The construction is based on the introduction of complex matrices, used in both encoding and decoding, which comprise sub-matrices of cascading connection values. The finite size effects are estimated for comparing the results to the bounds set by Shannon. The critical noise level achieved for certain code-rates and infinitely large systems nearly saturates the bounds set by Shannon even when the connectivity used is low.