On the Parity-Check Density and Achievable Rates of LDPC Codes

TL;DR

This paper derives new bounds relating the sparsity of parity-check matrices to the achievable rates of LDPC codes, improving understanding of their capacity and decoding thresholds.

Contribution

It introduces tighter bounds on parity-check density and achievable rates, advancing the theoretical limits for LDPC codes over symmetric channels.

Findings

New bounds on parity-check density as a function of capacity gap

Tighter upper bounds on achievable rates under ML decoding

Numerical examples demonstrating improved bounds for LDPC codes

Abstract

The paper introduces new bounds on the asymptotic density of parity-check matrices and the achievable rates under ML decoding of binary linear block codes transmitted over memoryless binary-input output-symmetric channels. The lower bounds on the parity-check density are expressed in terms of the gap between the channel capacity and the rate of the codes for which reliable communication is achievable, and are valid for every sequence of binary linear block codes. The bounds address the question, previously considered by Sason and Urbanke, of how sparse can parity-check matrices of binary linear block codes be as a function of the gap to capacity. The new upper bounds on the achievable rates of binary linear block codes tighten previously reported bounds by Burshtein et al., and therefore enable to obtain tighter upper bounds on the thresholds of sequences of binary linear block codes under ML decoding. The bounds are applied to low-density parity-check (LDPC) codes, and the improvement in their tightness is exemplified numerically.