Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation

TL;DR

This paper presents an efficient method to analyze the spectral shape of irregular doubly-generalized LDPC codes, providing exact evaluations and revealing symmetry properties in special cases, with practical implications for code design.

Contribution

It introduces a numerical technique for exact spectral shape evaluation of D-GLDPC codes and explores symmetry properties in specific code ensembles.

Findings

Efficient 4x4 polynomial system for spectral shape evaluation

Spectral shape simplifies for codes with uniform repetition variable nodes

Symmetry properties in local weight distributions induce spectral symmetry

Abstract

This paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient numerical technique for its evaluation is presented, involving the solution of a 4 x 4 system of polynomial equations. The expression is consistent with previous results, including the case where the normalized weight or stopping set size tends to zero. The spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree, a case which includes Tanner codes; for this case it is also shown how certain symmetry properties of the local weight distribution at the CNs induce a symmetry in the overall weight spectral shape function. Finally, using these new results, weight and stopping set size spectral shapes are evaluated for some example generalized and doubly-generalized LDPC code ensembles.