Single Parity Check Research Articles

Single Parity Check 부호를 적용한 3차원 Turbo Product 부호의 효율적인 복호 알고리즘

본 논문에서는 single parity check 부호(SPC)를 포함하는 3차원 turbo product 부호(TPC)의 효율적인 복호 기법을 제안한다. 일반적으로 TPC의 부호율을 극대화하기 위한 목적으로 부호 길이가 짧은 축에서 SPC 부호를 적용한다. 그러나 SPC 부호가 오류 정정 능력이 없는 부호이기 때문에 3차원 TPC를 Chase-Pyndiah 복호 알고리즘만으로 복호할 경우, 2차원 TPC에 비하여 성능 개선이 거의 발생하지 않는다. 본 논문에서는 이를 개선하기 위해 다음의 2가지 기법을 복호 과정에 적용하였다. 우선 SPC 부호로 이루어진 축에서는 구현 복잡도를 낮추기 위하여 <TEX>$min^*$</TEX>-sum 알고리즘을 복호 방법으로 적용하였으며, 반복 복호 방식으로는 성능 개선을 위해 직렬 복호 방식을 변형한 방식을 이용하였다. 마지막으로 이를 적용한 TPC 시뮬레이터의 성능을 비교 분석하고, 실제 하드웨어 구현과정에서 고려해야 할 부분을 소개한 후, VHDL을 이용하여 3차원 TPC를 설계하였다. In this paper, we propose a decoding scheme that can apply to a three dimensional turbo product code(TPC) with a single parity check code(SPC). In general, SPC is used an axis with shortest code length in order to maximize a code rate of the TPC. However, SPC does not have any error correcting capability, therefore, the error correcting capability of the three-dimensional TPC results in little improvement in comparison with the two-dimensional TPC. We propose two schemes to improve performance of three dimensional TPC decoder. One is <TEX>$min^*$</TEX>-sum algorithm that has advantages for low complexity implementation compared to Chase-Pyndiah algorithm. The other is a modified serial iterative decoding scheme for high performance. In addition, the simulation results for the proposed scheme are shown and compared with the conventional scheme. Finally, we introduce some practical considerations for hardware implementation.

An approach to the performance of SPC product codes on the erasure channel

Product codes can be used to correct errors or recover erasures.In this work we consider the simplest form of a product code, this is, the single parity check (SPC) product code.This code has a minimum distance of four and is thus guaranteed to recover all single,double, and triple erasure patterns.The code is actually capable of recovering a higher number of erasure patterns.We count the number of uncorrectable erasure patterns of size $n \times n $ with $t$ erasures, for $t=8$, $2n-3$, $2n-2$ and $ 2n-1$,using the relation between erasure patterns and bipartite graphs.

New decoding scheme for LDPC codes based on simple product code structure

In this paper, a new decoding scheme for low-density parity-check (LDPC) codes using the concept of simple product code structure is proposed based on combining two independently received soft-decision data for the same codeword. LDPC codes act as horizontal codes of the product codes and simple algebraic codes are used as vertical codes to help decoding of the LDPC codes. The decoding capability of the proposed decoding scheme is defined and analyzed using the paritycheck matrices of vertical codes and especially the combined-decodability is derived for the case of single parity-check (SPC) and Hamming codes being used as vertical codes. It is also shown that the proposed decoding scheme achieves much better error-correcting capability in high signal to noise ratio (SNR) region with low additional decoding complexity, compared with a conventional decoding scheme.

Single parity check-coded 16QAM over spatial superchannels in multicore fiber transmission.

We experimentally investigate single-parity check (SPC) coded spatial superchannels based on polarization-multiplexed 16-ary quadrature amplitude modulation (PM-16QAM) for multicore fiber transmission systems, using a 7-core fiber. We investigate SPC over 1, 2, 4, 5 or 7 cores in a back-to-back configuration and compare the sensitivity to uncoded PM-16QAM, showing that at symbol rates of 20 Gbaud and at a bit-error-rate (BER) of 10-3, the SPC superchannels exhibit sensitivity improvements of 2.7 dB, 2.0 dB, 1.7 dB, 1.3 dB, and 1.1 dB, respectively. We perform both single channel and wavelength division multiplexed (WDM) transmission experiments with 22 GHz channel spacing and 20 Gbaud channel symbol rate for SPC over 1, 3 and 7 cores and compare the results to PM-16QAM with the same spacing and symbol rate. We show that in WDM signals, SPC over hl1 core can achieve more than double the transmission distance compared to PM-16QAM at the cost of 0.91 bit/s/Hz/core in spectral efficiency (SE). When sharing the parity-bit over 7 cores, the loss in SE becomes only 0.13 bit/s/Hz/core while the increase in transmission reach over PM-16QAM is 44 %.

Block Markov Superposition Transmission of Repetition and Single-Parity-Check Codes

This letter presents a method to construct a family of codes with rate K/N(K = 1,2,...,N - 1) for any given integer N > 1. This family of codes, referred to as RSPC codes for short, consist of the repetition (R) code, the single-parity-check (SPC) code and the codes obtained by time-sharing between the R code and the SPC code. The RSPC codes have extremely simple encoding and soft-input soft-output (SISO) decoding algorithms, hence can be integrated into the recently proposed block Markov superposition transmission (BMST) system. The BMST introduces memory across short codes by spatially coupling the generator matrices of the short codes. A distinguished feature of the BMST system is the simple relation between the asymptotic coding gain and the spatially coupling depth (encoding memory). Furthermore, the BMST-RSPC codes can share a pair of easily reconfigurable encoder and decoder. Simulation results show that the BMST-RSPC codes perform well (within one dB away from the corresponding Shannon limits) in a wide range of code rates.

Asymmetric Slepian-Wolf Coding of Nonstationarily-Correlated M-ary Sources with Sliding-Window Belief Propagation

Asymmetric Slepian-Wolf coding (ASWC) of M-ary sources with nonstationary correlation is a very useful model for many practical problems. An effective implementation of this coding scheme is to binarize each M-ary source into multiple bitplanes which are then compressed by a single binary low-density parity-check (LDPC) code. Though the inter-bitplane correlation of M-ary sources can be exploited at the decoder by the joint-bitplane belief propagation (JBBP) algorithm, accurate online estimation of varying local source correlation still remains a major challenge. To tackle this problem, this paper proposes an M-ary counterpart of the sliding-window belief propagation (SWBP) algorithm to realize simultaneous source recovery and correlation estimation. To search for the optimal sliding-window size, the expected rate is raised as a new criterion. Moreover, an adaptive method is proposed to decide whether correlation reestimation is necessary. The M-ary SWBP (MSWBP) algorithm is then generalized to obtain its 2D form, which can be used to tackle the ASWC of 2D M-ary sources with nonstationary correlation. The developed 1D/2D-MSWBP algorithm inherits all merits of the original binary SWBP algorithm, e.g., near-optimal coding efficiency, low complexity, insensitivity to initial settings, etc., making it a very attractive technique in practice.

SPC-APPM coded modulation for deep-space optical communications

A coded modulation scheme for deep-space optical communications is proposed, which is composed of an outer single-parity-check (SPC)-based product code, an interleaver, a bit-accumulator and a pulse-position modulation (PPM). It is referred as SPC-APPM code, which is decoded with an iterative demodulator-decoder using standard turbo-decoding techniques. Investigations show that the scheme has the advantages of low encoding and decoding complexities, good performance and flexible code rate for all rates above 1/2. Meanwhile, simulation results demonstrate that the SPC-APPM provides the performance similar to the low-density parity-check-APPM (LDPC-APPM), superior to the LDPC-PPM and product accumulate code-PPM (PA-PPM), although inferior to serially concatenated PPM (SCPPM). At the bit error rate (BER) of 10−5, the performance of SPC-APPM is about 0.7 dB better than LDPC-PPM and 1.2 dB better than PA-PPM.

Complexity Reduction of Ordered Statistics Decoding Using Side Information

In this letter, the complexity of the ordered statistics decoding of linear binary block codes is reduced assuming that partial knowledge about the transmitted codeword is available at the decoder. Such side information is provided to the decoder at the receiver from the encoder at the transmitter over a highly reliable low-rate channel. Side information is exploited at the decoder to significantly reduce the number of test codewords that have to be examined during the decoding without sacrificing the error rate performance of the decoder. Side information is represented either by the single parity checks of non-overlapping codeword segments, or by the Hamming weight of the codeword or its part. The proposed method can be readily integrated with other list decoding techniques. Finally, implementation of the proposed complexity reducing scheme in practical systems is also discussed.

Extended Single Parity Check Product Codes that Achieve Close-to-Capacity Performance in High Coding Rate

This letter proposes extended single parity check product codes and presents their empirical performances on a Gaussian channel by belief propagation (BP) decoding algorithm. The simulation results show that the codes can achieve close-to-capacity performance in high coding rate. The code of length 9603 and of rate 0.96 is only 0.77dB away from the Shannon limit for a BER of 10-5.

Enhanced Upper Bound for Erasure Recovery in SPC Product Codes

Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a closed-form tight upper bound for the post-decoding erasure rate. Closed-form exact expressions are derived for up to seven erasures. Previously published closed-form bounds assumed that all unrecoverable patterns should contain four erasures in a square. Additional non-square patterns are accounted for in the proposed expressions. The derived expressions are verified using exhaustive search. Eight or more erasures are accounted for by using a bound. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or search algorithms, whether exhaustive or novel.

LDPC codes based on serially concatenated multiple parity-check codes

This letter proposes a new class of serially concatenated codes that can be viewed as low-density parity- check (LDPC) codes. They are derived from multiple serially concatenated single parity-check (M-SC-SPC) codes, but they use different components, that we call multiple parity-check (MPC) codes. In comparison with M-SC-SPC codes, the new scheme achieves better performance with similar complexity. The proposed codes can represent an alternative to the well-known family of repeat accumulate (RA) codes, being based on the same principles.

Performance Comparison between Turbo Code and Rate-Compatible LDPC Code for Evolved UTRA Downlink OFDM Radio Access

This paper compares the turbo code and rate-compatible low-density parity-check (LDPC) codes based on the block error rate (BLER) performance and decoding complexity in order to clarify which channel coding scheme is most appropriate for the channel coding scheme in the OFDM based Evolved UTRA (E-UTRA) downlink. Simulation results and the decoding complexity analysis show that although the Rate-Compatible/Quasi-Cyclic (RC/QC)-LDPC code employing an offset layered belief propagation (BP) method can reduce the computational complexity by approximately 30% for the channel coding rate of R ≥ 1/2, the required average received signal energy per bit-to-noise power spectrum density ratio (Eb/N0) is degraded by approximately 0.2-0.3dB for R=1/3, 1/2 and 3/4 compared to that for the turbo code. Moreover, the decoding complexity level of the RC/QC-LDPC code with the δ-min algorithm is almost the same or higher than that for the turbo code with a slight degradation in the required received Eb/N0. Although the decoding complexity level of the ZigZag code is lower than that of the turbo code, the code brings about a distinct loss in the required average received Eb/N0 of approximately 0.4dB. Finally, the turbo Single Parity Check (SPC) code improves the BLER performance compared to the ZigZag code, i.e., achieves almost the same BLER performance as that for the turbo code, at the cost of a two-fold increase in the decoding complexity. As a result, we conclude that the turbo code with a contention free interleaver is more promising than the LDPC codes for prioritizing the achievable performance over complexity and as the channel coding scheme for the shared data channel in the E-UTRA.

Iterative Coded Modulation With Code Rate Flexibility for Optical Space Communications

We propose an iterative coded modulation technique with code rate flexibility for optical space communications. It is a modified version of the existing SCPPM, which consists of the serial concatenation of an outer convolutional code, an interleaver, a bit-accumulator, and pulse-position modulation (PPM), and has been demonstrated to give the best performance on an optical link modeled as PPM/Poisson channels. The primary modification is to add a single-parity-check (SPC) code. We refer to the coded modulation as SPC-SCPPM. The two codes are decoded with an iterative demodulator-decoder, using standard turbo-decoding techniques. We compare the two coded modulation techniques on a PPM/Poisson channel. The rate-1/2 SPC-SCPPM provides performance similar to the SCPPM of the same parameters. However, SPC-SCPPM has been shown to possess code rate flexibility for all rates higher than 1/2 in contrast to SCPPM.

Iterative Decoding of SPC Outer Coded Concatenation Codes with Maximal Ratio Combining

This letter proposes a simple iterative decoding algorithm for the concatenation codes where the outer code is single-parity-check (SPC) code. The erroneous inner codewords are iteratively combined with maximum ratio combining (MRC) and then re-decoded. Compared with the conventional scheme where the RS outer code concatenation is algebraically decoded to recover the erasures, the proposed scheme has better performance due to MRC processing. On the other hand, the proposed scheme is less complex because the linear combination is simpler than algebraical decoding and the MRC gain can loose the requirement for inner decoder.

A New Linear Programming Approach to Decoding Linear Block Codes

In this paper, we propose a new linear programming formulation for the decoding of general linear block codes. Different from the original formulation given by Feldman, the number of total variables to characterize a parity-check constraint in our formulation is less than twice the degree of the corresponding check node. The equivalence between our new formulation and the original formulation is proven. The new formulation facilitates to characterize the structure of linear block codes, and leads to new decoding algorithms. In particular, we show that any fundamental polytope is simply the intersection of a group of the so-called minimum polytopes, and this simplified formulation allows us to formulate the problem of calculating the minimum Hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables. We then propose a branch-and-bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem. In addition, we prove that, for the family of single parity-check (SPC) product codes, the fractional distance and the pseudodistance are both equal to the minimum distance. Finally, we propose an efficient algorithm for decoding SPC product codes with low complexity and maximum-likelihood (ML) decoding performance.

Generalized LDPC codes and generalized stopping sets

A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.

On the Concatenation of Soft Inner Code With Reed–Solomon Code for Perpendicular Magnetic Recording

We evaluate the performance of coding system consisting of inner code and Reed-Solomon (RS) code. Three types of inner codes are considered: single-parity check (SPC) code, multiparity check code, and interleaved single-parity check (ISPC) code. The evaluation is performed by simulation on a perpendicular recording channel model. Evaluation methods include direct counting of bit errors and sector errors, and semianalytic method based on block multinomial model. The performance of the coding system under detectable and undetectable media defects is also studied

Simplified LLRs for the Decoding of Single Parity Check Turbo Product Codes Transmitted Using 16QAM

Iterative soft-decision decoding algorithms require channel log-likelihood ratios (LLRs) which, when using 16QAM modulation, require intensive computations to be obtained. Therefore, we derive four simple approximate LLR expressions. When using the maximum a posteriori probability algorithm for decoding single parity check turbo product codes (SPC/TPCs), these LLRs can be simplified even further. We show through computer simulations that the bit-error-rate performance of(8,7)2and(8,7)3SPC/TPCs, transmitted using 16QAM and decoded using the maximum a posteriori algorithm with our simplified LLRs, is nearly identical to the one achieved by using the exact LLRs.

Performance analysis of fiber-optic BPPM CDMA systems with single parity-check product codes

The application of a family of single parity-check (SPC) product codes in the fiber-optic code-division multiple-access (CDMA) systems with binary pulse position modulation (BPPM) is considered in this paper. We evaluate the bit-error probabilities of the coded systems, taking into account the photodetector noise, thermal noise, and the multiple-access interference. In this paper, the performance of the employed coding schemes over a binary-input output-symmetric channel is analyzed, based on the density evolution approach. The validity of the analytically obtained bit-error rates (BERs) over binary symmetric channels (BSCs) for each of the decoding iterations is verified by the system simulation. It is then shown that the application of these coding schemes in the optical BPPM-CDMA communication system, which can be modeled as a BSC, significantly improves the system performance. Particularly, for a certain BER (less than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-7</sup> ) and bandwidth, the coded systems not only permit a higher number (more than twice) of active users, compared with the uncoded systems, but also can operate at higher channel bit rate (up to four times) with more than 10% energy saving. Furthermore, by increasing the number of code dimensions, less power (less than 40%) is required to achieve the desired performance

Analysis of low-density parity-check codes based on EXIT functions

We exploit extrinsic information transfer functions of single parity-check and repetition codes over the binary input additive white Gaussian noise (biAWGN) channel, derived by the authors, for asymptotic performance analysis of belief propagation decoding of low-density parity-check codes. The approach is based on a Gaussian approximation (GA) of the density evolution algorithm using the mutual information measure. We show that this method allows more accurate prediction of the decoding threshold in the biAWGN channel than the earlier known GA methods