On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes

TL;DR

This paper derives a general expression for the asymptotic growth rate of small linear-weight codewords in irregular D-GLDPC codes, linking code structure to growth behavior and extending stability analysis to these codes.

Contribution

It provides a compact, generalized formula for the growth rate of small codewords in irregular D-GLDPC codes, extending previous LDPC results and connecting growth behavior to stability conditions.

Findings

Ensembles with check or variable node minimum distance > 2 have good growth rate behavior.

A key parameter determines whether the expected number of small codewords is small or large.

Growth rate depends only on nodes with minimum distance 2 in certain cases.

Abstract

In this paper, an expression for the asymptotic growth rate of the number of small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or variable node minimum distance greater than 2 are shown to be have good growth rate behavior, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular D-GLDPC codes.