Improved Rate-versus-Distance Upper Bounds for LDPC Codes

TL;DR

This paper introduces a new framework using coset-weight generating functions to improve upper bounds on the rate of LDPC codes for various relative distances.

Contribution

It develops a novel approach for estimating coset ball sizes, sharpening previous bounds and enhancing understanding of LDPC code rate-distance tradeoffs.

Findings

Sharpened upper bounds on LDPC code rates for certain distances.

Introduced coset-weight generating functions for better analysis.

Improved bounds using local growth analysis of low-weight vectors.

Abstract

LDPC codes play a vital role in coding theory and practical error correction. A central problem in this direction is to understand their rate--distance tradeoff. In this paper, we introduce a new framework for estimating ball sizes in the coset graphs of LDPC codes. The key new object is the coset-weight generating function, which encodes the minimum Hamming weights of all cosets of a linear code. Rather than estimating coset balls directly, we upper-bound this generating function through a local growth analysis for codes spanned by low-weight vectors. This framework sharpens the previous ball-size estimate of Iceland and Samorodnitsky. Combined with a general method of Friedman and Tillich that relates balls in coset graphs to sizes of error-correcting codes, it further improves the upper bounds on the rate of LDPC codes for a significant range of relative distances.