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IEEE International Symposium on Information Theory (ISIT)

Edge spreading design of high rate array-based SC-LDPC codes

David G. M. Mitchell, Eirik Rosnes

Published Jun 1, 2017• 27 citations
Publisher

Abstract

Absorbing sets (ASs) are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we propose an edge spreading approach to construct high rate array-based spatially-coupled LDPC codes by jointly optimizing the AS spectrum and the minimum distance. By considering general edge spreadings and by considering a larger memory, we show that strictly better codes can be constructed, both in terms of achievable minimum distance for small-to-moderate block lengths and in terms of the number of small ASs.

DOI
10.1109/isit.2017.8007068

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Absorbing sets (ASs) are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we propose an edge spreading approach to construct high rate array-based spatially-coupled LDPC codes by jointly optimizing the AS spectrum and the minimum distance. By considering general edge spreadings and by considering a larger memory, we show that strictly better codes can be constructed, both in terms of achievable minimum distance for small-to-moderate block lengths and in terms of the number of small ASs.
Absorbing sets are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we study the absorbing set properties of a class of high-rate array-based spatially coupled LDPC (SC-LDPC) codes that are constructed by coupling together L array-based LDPC block codes. We prove that the smallest absorbing sets existing in the Tanner graph of the SC-LDPC code have the same size as those in the corresponding uncoupled LDPC codes, and the number of such sets grow linearly with L. We show that spatial coupling greatly reduces the average number (per symbol) of minimal sets compared to the uncoupled codes, and we explain that this reduction is due to many absorbing sets and small cycles being `broken' by the coupling process. The large reduction in the number of minimal absorbing sets suggests that array-based SC-LDPC codes will have significantly improved decoding performance in the high signal-to-noise ratio regime compared to the corresponding uncoupled LDPC codes.
Absorbing sets are known to be the primary factor in the error-floor performance of low-density parity-check (LDPC) codes with message passing decoders over the additive white Gaussian noise (AWGN) channel. Besides showing excellent waterfall performance, spatially coupled LDPC (SC-LDPC) codes that are constructed by an edge spreading technique are known to have fewer cycles and absorbing sets than their block code counterparts, and therefore to exhibit better error-floor performance. Based on our previously obtained results for quantized LDPC block decoders, we derive lower bounds on the performance of quantized SC-LDPC decoders, including both a flooding schedule decoder and a sliding window decoder. Numerical simulation results confirm the accuracy of the obtained bounds and show that, for quantized decoders, properly designed SC-LDPC codes have better error-floor performance than their underlying LDPC block codes.
In the infinite blocklength regime, spatially-coupled LDPC codes are capable of achieving capacity-approaching performance under message-passing decoding. In the finite blocklength regime, it is known that absorbing sets compete with the codewords to be the output of sub-optimal message-passing decoders: the existence of such sets in the Tanner graph of LDPC codes causes performance degradation in the low error rate region. This paper presents a mathematical approach to finding the exact number of absorbing sets in array-based spatially-coupled (AB-SC) codes. Our analysis is universal in the sense that it is in principle applicable to absorbing sets of any size. Moreover, all design parameters of AB-SC codes such as the coupling length, the circulant size, and the cutting vector are considered in the presented count. Based on our analysis, we present an approach to find provably minimal cutting vectors, with respect to the number of absorbing sets, for the construction of AB-SC codes with various circulant sizes. Simulation results show the superior error floor performance of AB-SC codes with the minimal cutting vector compared to AB-SC codes with randomly-selected cutting vectors. We also provide the average number of non-binary absorbing sets in the Tanner graph of non-binary AB-SC codes constructed by uninformed (random) assignment of edge weights to a binary AB-SC code.
Abstract It is well known that some harmful objects in the Tanner graph of low-density parity-check (LDPC) codes have a negative impact on their error correction performance under iterative message-passing decoding. Depending on the channel and the decoding algorithm, these harmful objects are different in nature and can be stopping sets, trapping sets, absorbing sets, or pseudocodewords. Differently from LDPC block codes, the design of spatially coupled LDPC codes must take into account the semi-infinite nature of the code, while still reducing the number of harmful objects as much as possible. We propose a general procedure, based on edge spreading, enabling the design of good quasi-cyclic spatially coupled LDPC (QC-SC-LDPC) codes. These codes are derived from quasi-cyclic LDPC (QC-LDPC) block codes and contain a considerably reduced number of harmful objects with respect to the original QC-LDPC block codes. We use an efficient way of enumerating harmful objects in QC-SC-LDPCCs to obtain a fast algorithm that spans the search space of potential candidates to select those minimizing the multiplicity of the target harmful objects. We validate the effectiveness of our method via numerical simulations, showing that the newly designed codes achieve better error rate performance than codes presented in previous literature.
The error correction performance of low-density parity-check (LDPC) codes under iterative message-passing decoding is degraded by the presence of certain harmful objects existing in their Tanner graph representation. Depending on the context, such harmful objects are known as stopping sets, trapping sets, absorbing sets, or pseudocodewords. In this paper, we propose a general procedure based on edge spreading that enables the design of quasi-cyclic (QC) spatially coupled low-density parity-check codes (SC-LDPCCs) that are derived from QC-LDPC block codes and possess a significantly reduced multiplicity of harmful objects with respect to the original QC-LDPC block code. This procedure relies on a novel algorithm that greedily spans the search space of potential candidates to reduce the multiplicity of the target harmful object(s) in the Tanner graph. The effectiveness of the method we propose is validated via examples and numerical computer simulations.
The error correction performance of low-density parity-check codes under iterative message-passing decoding is degraded by the presence of certain harmful objects existing in their Tanner graph representation. Depending on the context, such harmful objects are known as stopping sets, trapping sets, absorbing sets, or pseudocodewords. In this paper, we propose a general procedure, based on edge spreading, that enables the design of good quasi-cyclic spatially coupled low-density parity-check codes. These codes are derived from quasi-cyclic low-density parity-check (QC-LDPC) block codes and possess a significantly reduced multiplicity of harmful objects with respect to the original QC-LDPC block codes. The proposed procedure relies on a novel algorithm that greedily spans the search space of potential candidates to reduce the multiplicity of the target harmful objects. The effectiveness of the method is validated via examples and numerical computer simulations.