Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

TL;DR

This paper analyzes the finite-length performance of spatially-coupled LDPC codes over burst-erasure channels, providing bounds, improvements through expurgation, and extensions to more complex erasure models, verified by simulations.

Contribution

It introduces a finite-length analysis framework for SC-LDPC codes on burst-erasure channels, including bounds, ensemble improvements, and extensions to multiple and random bursts.

Findings

Tight lower bounds for block erasure probability at finite length.

Expurgation significantly improves burst erasure correction.

Extended analysis to multiple and random burst models.

Abstract

Regular spatially-Coupled LDPC (SC-LDPC) ensembles have gained significant interest since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this work, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per codeword and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability ($P_{\mathrm{B}}$) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random position in the codeword and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximnum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability $p$, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.