Spectral Shape of Check-Hybrid GLDPC Codes

TL;DR

This paper derives a simple expression for the asymptotic spectral shape of hybrid GLDPC codes, revealing how local symmetries influence the overall weight and stopping set size spectra.

Contribution

It provides a unified analytical expression for the spectral shape of hybrid GLDPC codes, extending previous results and highlighting symmetry effects.

Findings

Derived a simple spectral shape expression for hybrid GLDPC codes

Confirmed consistency with existing results for small weight/stopping set sizes

Identified symmetry properties affecting the spectral shape function

Abstract

This paper analyzes the asymptotic exponent of both the weight spectrum and the stopping set size spectrum for a class of generalized low-density parity-check (GLDPC) codes. Specifically, all variable nodes (VNs) are assumed to have the same degree (regular VN set), while the check node (CN) set is assumed to be composed of a mixture of different linear block codes (hybrid CN set). A simple expression for the exponent (which is also referred to as the growth rate or the spectral shape) is developed. This expression is consistent with previous results, including the case where the normalized weight or stopping set size tends to zero. Furthermore, it is shown how certain symmetry properties of the local weight distribution at the CNs induce a symmetry in the overall weight spectral shape function.