Exact Free Distance and Trapping Set Growth Rates for LDPC Convolutional Codes

TL;DR

This paper analyzes the free distance and trapping set growth rates of LDPC convolutional codes using protograph-based methods, providing bounds that help understand their error-correcting capabilities and structural properties.

Contribution

It introduces a new approach to bound the free distance and trapping set sizes of periodically time-varying LDPC convolutional codes using protograph analysis.

Findings

Bounds on free distance growth rates are tight for large periods.

The method extends to trapping set size bounds.

Ensembles are shown to be asymptotically good.

Abstract

Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.