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IEEE Communications Letters

Irregular LDPC codes as concatenation of regular LDPC codes

M.P.C. Fossorier, N. Miladinovic

Published Jul 1, 2005• 10 citations
Publisher

Abstract

In this letter, the construction of irregular LDPC codes obtained by concatenating regular ones is considered. These codes are analyzed for the binary symmetric channel (BSC). It is shown via proper distribution evolution that such concatenated codes have in general a threshold value weaker (but not necessarily much different) than their unconstrained counterparts for the BSC. On the other hand, they can offer advantages in term of convergence and error performance for lengths of practical interest and long enough to validate the degree distribution selection.

DOI
10.1109/lcomm.2005.1461686

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In this letter, the construction of irregular LDPC codes obtained by concatenating regular ones is considered. These codes are analyzed for the binary symmetric channel (BSC). It is shown via proper distribution evolution that such concatenated codes have in general a threshold value weaker (but not necessarily much different) than their unconstrained counterparts for the BSC. On the other hand, they can offer advantages in term of convergence and error performance for lengths of practical interest and long enough to validate the degree distribution selection.
Decoding performance of LDPC (Low-Density Parity-Check) codes is highly dependent on the degree distributions of the Tanner graphs which define the LDPC codes. We compare two LDPC code ensembles, one has a uniform degree distribution and the other a non-uniform one over a BEC (Binary Erasure Channel) and a BSC (Binary Symmetric Channel) thorough DE (Density Evolution). We then derive sufficient conditions on the erasure probability of a BEC and the error probability of a BSC, under which the LDPC code ensembles with uniform degree distributions outperform those with non-uniform degree distributions.
LDPC code shows a good performance with long-block codes. However, certain channels are constrained to use short-block codes due to latency. Therefore, concatenated LDPC codes with iterative decoding is a good choice to get a good performance. Concatenated binary LDPC codes were introduced as a class of concatenated codes in which the LDPC codes are irregular codes having different parameters. Although irregular LDPC codes are more efficient than regular codes, irregular LDPC codes have an error floor and a higher encoding complexity than regular code. In this paper, in order to get a good performance/complexity trade-off with a short-block code, we investigate a parallel concatenation of two identical regular binary LDPC codes, using an interleaver. Simulation results show that the proposed code outperforms a single LDPC code. The proposed code needs less time decoding delay than a single LDPC code with the same iteration number. Also, the proposed code needs a less number of iterations to achieve the same performance of a single LDPC code that leads to less decoding complexity and decoding delay. A simplified algorithm, the min-sum algorithm, that is used to decode the component codes shows a small performance loss with respect to the sum-product algorithm.
High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.
Low-density parity-check (LDPC) codes can approach the Shannon limit performance closely, and are becoming one of the most promising channel codes in the Error Control Coding area.The performance of an LDPC code with given length is mainly affected by the sizes of some combinatorial characteristics of its corresponding bipartite graph and the distribution of variable degrees and check node degrees.With the help of the density evolution algorithm, randomly designed LDPC codes, with carefully chosen degree distribution pairs, have been shown to achieve better Shannon capacity performance than their regular counterparts when decoded using the iterative belief propagation (BP) decoding algorithm.However, regular LDPC codes outperform their irregular counterparts in term of their error floors.Therefore, construction of LDPC codes which have both attractive error rate performance and low error floor performance has become an attractive topic.Currently, optimization of variable node degree distribution and check node degree distribution for an LDPC code with given length, which could help the code to achieve good Shannon limit performance, has been extensively studied.In this dissertation, we demonstrate some approaches of improving the error floor performance for a given LDPC code with desired degree distribution pair.The contribution mainly includes three parts.In the first part, the relationship between cycles and unfavorable combinatorial
In this paper we design non-uniform bit-wise puncturing distributions for irregular non-binary LDPC (NB-LDPC) codes. The puncturing distributions are optimized by minimizing the decoding threshold of the punctured LDPC code, the threshold being computed with a Monte-Carlo implementation of Density Evolution. First, we show that Density Evolution computed with Monte-Carlo simulations provides accurate (very close) and precise (small variance) estimates of NB-LDPC code ensemble thresholds. Based on the proposed method, we analyze several puncturing distributions for regular and semi-regular codes, obtained either by clustering punctured bits, or spreading them over the symbol-nodes of the Tanner graph. Finally, optimized puncturing distributions for non-binary LDPC codes with small maximum degree are presented, which exhibit a gap between 0.2 and 0.5 dB to the channel capacity, for punctured rates varying from 0.5 to 0.9.
In this work, we analytically derive the evolution of single-bit messages in the bit-flipping algorithm (BFA) for irregular low-density parity-check (LDPC) codes on the binary symmetric channel (BSC). In particular, the analysis tracks the probability distributions of the flipping functions and the bit-decision error at each iteration, which is possible by introducing a state variable at each variable node. The probability distributions are indirectly obtained by taking advantage of the state and state transition probabilities. The analysis will be confirmed by comparing analytic evaluations with simulation results. For the comparison, we implement LDPC codes at a finite length.