Improved Rate-versus-Distance Upper Bounds for LDPC Codes

2026-05-01

AI Summary

This work investigates the trade-off between code rate and relative distance for LDPC codes, aiming to tighten the theoretical upper bound on achievable rates. To this end, the authors introduce a novel analytical framework based on the coset weight generating function, which integrates local growth analysis with the Friedman–Tillich approach. Rather than directly estimating the size of balls in the coset graph, the method achieves a more precise upper bound by characterizing the structural properties of cosets of linear codes. The resulting bound significantly improves upon existing results across a wide range of relative distances and surpasses the classical bound established by Iceland and Samorodnitsky, thereby offering a tighter theoretical characterization of the fundamental performance limits of LDPC codes.

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.

Problem

• LDPC codes
• rate--distance tradeoff
• upper bounds
• coset graphs
• error-correcting codes

Innovation

• LDPC codes
• coset-weight generating function
• rate--distance tradeoff
• ball-size estimation
• local growth analysis