Free Pseudodistance Growth Rates for Spatially Coupled LDPC Codes over the BEC

TL;DR

This paper establishes that the minimum pseudoweight of certain spatially coupled LDPC codes grows linearly with code length, providing bounds and exact growth rates for decoding performance over the BEC.

Contribution

It introduces bounds on the free pseudodistance growth rate for SC-LDPC codes using tail-biting and terminated ensembles, and shows these bounds coincide for large periods.

Findings

Minimum pseudoweight grows linearly with code length.

Bounds on pseudodistance growth rate are tight for large periods.

Exact growth rate can be determined for specific code ensembles.

Abstract

The minimum pseudoweight is an important parameter related to the decoding performance of LDPC codes with iterative message-passing decoding. In this paper, we consider ensembles of periodically time-varying spatially coupled LDPC (SC-LDPC) codes and the pseudocodewords arising from their finite graph covers of a fixed degree. We show that for certain $(J,K)$-regular SC-LDPC code ensembles and a fixed cover degree, the typical minimum pseudoweight of the unterminated (and associated tail-biting/terminated) SC-LDPC code ensembles grows linearly with the constraint (block) length as the constraint (block) length tends to infinity. We prove that one can bound the the free pseudodistance growth rate over a BEC from below (respectively, above) using the associated tail-biting (terminated) SC-LDPC code ensemble and show empirically that these bounds coincide for a sufficiently large period, which gives the exact free pseudodistance growth rate for the SC-LDPC ensemble considered.