Randomly Punctured LDPC Codes

Abstract

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binaryinput additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results. Index Terms-Low-density parity-check (LDPC) codes, spatially coupled codes, rate-compatible codes, punctured codes, iterative decoding, belief propagation, decoding thresholds, binary erasure channel, additive white Gaussian noise channel, minimum distance.

Spatially Coupled Turbo-Like Codes: A New Trade-Off Between Waterfall and Error Floor

“…We consider the same TC ensembles as those in [9]- [11], namely parallel concatenated codes (PCCs) [1], serially concatenated codes (SCCs) [12], [13], braided convolutional codes (BCCs) [14], [15], and hybrid concatenated codes (HCCs) [16], [17]. As the first step of our investigation, using the decoding thresholds of the binary erasure channel (BEC) obtained in [10], [11] and the method described in [18], [19], we predict the decoding thresholds over the additive white Gaussian noise (AWGN) channel. Using these thresholds together with the provided simulation results, we discuss the effect of spatial coupling on the performance of TCs in the waterfall region over the AWGN channel.…”

Thresholds of Braided Convolutional Codes on the AWGN Channel

“…The results of the mixed predictions are shown in Fig. 4 for However, the estimated thresholds do not strictly lie on a straight line as was the case in [6]. It can be seen in Table II that for SC-BCCs, the MC-DE and predicted thresholds are almost identical.…”

“…As an efficient alternative, we then consider the erasure channel approximation by Chung [9] for predicting the AWGN channel thresholds from those of the BEC and compare the results. Finally, we demonstrate that for randomly punctured ensembles, analogously to LDPC codes [6], the thresholds of BCCs of all higher rates can immediately be predicted from the unpunctured thresholds of the BEC and/or the AWGN channel. Some simulation results are also given.…”

“…Can we predict the thresholds such that the difference of the predicted and the estimated threshold do not increase as the fraction of puncturing increases? In [6], it has been shown that for the P2 punctured LDPC codes, it is possible to closely predict the thresholds on the AWGN channel, given just the threshold of the unpunctured code ensemble and the design rate R. In the following, we will apply the methods discussed in [6] to the randomly punctured BCC ensembles and discuss the results obtained by it.…”

On LDPC code ensembles with generalized constraints

“…In this regard, broader classes of component codes or improved decoding methods at GC nodes could be incorporated in a systematic way. In addition, one of our main lines of research is the extension of the proposed analysis of GLDPC ensembles to incorporate rate adaptation techniques by means of puncturing [22] and the use of generalized variable nodes [23]. Rate=0.1667, Gap=0.0823 Fig.…”