The Cycle Consistency Matrix Approach to Absorbing Sets in Separable Circulant-Based LDPC Codes

TL;DR

This paper introduces the cycle consistency matrix (CCM) as an analytical tool to characterize and avoid absorbing sets in separable circulant-based LDPC codes, leading to improved error-floor performance under limited-precision decoding.

Contribution

The paper presents the CCM method for analyzing and constructing SCB LDPC codes to eliminate small absorbing sets, especially in high-rate scenarios.

Findings

CCM analysis determines the presence and multiplicity of absorbing sets.

CCB-based constructions can completely avoid certain small absorbing sets.

Simulation shows improved error-floor slopes and up to tenfold FER reduction.

Abstract

For LDPC codes operating over additive white Gaussian noise channels and decoded using message-passing decoders with limited precision, absorbing sets have been shown to be a key factor in error floor behavior. Focusing on this scenario, this paper introduces the cycle consistency matrix (CCM) as a powerful analytical tool for characterizing and avoiding absorbing sets in separable circulant-based (SCB) LDPC codes. SCB codes include a wide variety of regular LDPC codes such as array-based LDPC codes as well as many common quasi-cyclic codes. As a consequence of its cycle structure, each potential absorbing set in an SCB LDPC code has a CCM, and an absorbing set can be present in an SCB LDPC code only if the associated CCM has a nontrivial null space. CCM-based analysis can determine the multiplicity of an absorbing set in an SCB code and CCM-based constructions avoid certain small absorbing sets completely. While these techniques can be applied to an SCB code of any rate, lower-rate SCB codes can usually avoid small absorbing sets because of their higher variable node degree. This paper focuses attention on the high-rate scenario in which the CCM constructions provide the most benefit. Simulation results demonstrate that under limited-precision decoding the new codes have steeper error-floor slopes and can provide one order of magnitude of improvement in the low FER region.