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Belief Propagation Decoding for Short-Length Codes Based on Sparse tanner graph

IEEE COMMUNICATIONS LETTERS 2024年第5期28卷 969-973页

作者： Li, Zongyao Shen, Yifei Ren, Yuqing Huang, Yongming You, Xiaohu Zhang, Chuan Southeast Univ Natl Mobile Commun Res Lab LEADS Nanjing 211189 Peoples R China Southeast Univ Frontiers Sci Ctr Mobile Informat Commun & Secur Nanjing 211189 Peoples R China Purple Mt Labs Nanjing 211100 Peoples R China Ecole Polytech Fed Lausanne EPFL Telecommun Circuits Lab TCL CH-1015 Lausanne Switzerland

In this letter, we present a novel approach to improve belief propagation (BP) decoding performance by sparsifying the tanner graph. Our proposed method first constructs a low-weight parity-check (LWPC) matrix with increased rows based on the original parity-check matrix. Then, all 4-cycles in the LWPC matrix are eliminated by adding auxiliary variable nodes and corresponding parity-check rows, resulting in the generalized LWPC (G-LWPC) matrix. Compared to the original matrix, the sparsity is greatly improved without altering the encoding constraints. Simulation results show that BP decoding based on the G-LWPC matrix is particularly effective for short code lengths. For a (128, 20) polar code with 24-bit cyclic redundancy check, our proposed G-LWPC matrix reduces the density from 17.4% to 1.3%, and with 10 iterations, BP decoding can achieve the performance of successive cancellation list decoding with a list size of 32.

关键词： Decoding Sparse matrices Polar codes Iterative decoding Standards Reliability Complexity theory Belief propagation tanner graph minimum-weight codeword low-weight parity-check sparse graph

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An Efficient Strategy to Count Cycles in the tanner graph of Quasi-Cyclic LDPC Codes

IEEE JOURNAL ON SELECTED AREAS IN INFORMATION THEORY

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IEEE JOURNAL ON SELECTED AREAS IN INFORMATION THEORY 2023年 4卷 499-513页

作者： Gomez-Fonseca, Anthony Smarandache, Roxana Mitchell, David G. M. Univ Notre Dame Dept Math Notre Dame IN 46556 USA Univ Notre Dame Dept Elect Engn Notre Dame IN 46556 USA New Mexico State Univ Klipsch Sch Elect & Comp Engn Las Cruces NM 88003 USA

In this paper, we present an efficient strategy to enumerate the number of k-cycles, g = 0. We provide formulas for the number of k-cycles, N-k, by just taking into account repetitions in some multisets constructed fr... 详细信息

In this paper, we present an efficient strategy to enumerate the number of k-cycles, g <= k < 2g, in the tanner graph of a quasi-cyclic low-density parity-check (QC-LDPC) code with girth g using its polynomial parity-check matrix H. This strategy works for both (d(v), d(c))-regular and irregular QC-LDPC codes. In this approach, we note that the mth power of the polynomial adjacency matrix can be used to describe walks of length m in the protograph and can therefore be sufficiently described by the matrices Bm(H) (sic) (HHT)([m/2]) H-(m mod 2), where m >= 0. We provide formulas for the number of k-cycles, N-k, by just taking into account repetitions in some multisets constructed from the matrices B-m(H). This approach is shown to have low complexity. For example, in the case of QC-LDPC codes based on the 3 x n(v) fully-connected protograph, the complexity of determining N-k, for k = 4, 6, 8, 10 and 12, is O(n(v)(2) log(N)), O(n(v)(2) log(nv) log(N)), O(n(v)(4) log(4)(nv) log(N)), O(n(v)(4) log(nv) log(N)) and O(n(v)(6) log(6)(n(v)) log(N)), respectively. The complexity, depending logarithmically on the lifting factor N, gives our approach, to the best of our knowledge, a significant advantage over previous works on the cycle distribution of QC-LDPC codes.

关键词： LDPC codes quasi-cyclic LDPC codes cycles girth tanner graph

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Low-loss belief propagation decoder with tanner graph in quantum error-correction codes

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Chinese Physics B 2022年第1期31卷 143-149页

作者： Dan-Dan Yan Xing-Kui Fan Zhen-Yu Chen Hong-Yang Ma School of Sciences Qingdao University of TechnologyQingdao 266033China

Quantum error-correction codes are immeasurable resources for quantum computing and quantum ***,the existing decoders are generally incapable of checking node duplication of belief propagation(BP)on quantum low-density parity check(QLDPC)*** on the probability theory in the machine learning,mathematical statistics and topological structure,a GF(4)(the Galois field is abbreviated as GF)augmented model BP decoder with tanner graph is *** problem of repeated check nodes can be solved by this *** simulation,when the random perturbation strength p=0.0115-0.0116 and number of attempts N=60-70,the highest decoding efficiency of the augmented model BP decoder is obtained,and the low-loss frame error rate(FER)decreases to 7.1975×10^(-5).Hence,we design a novel augmented model decoder to compare the relationship between GF(2)and GF(4)for quantum code[[450,200]]on the depolarization *** can be verified that the proposed decoder provides the widely application range,and the decoding performance is better in QLDPC codes.

关键词： tanner graph belief propagation decoder augmented model fourier transform

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Detecting Cycles of Length 8 in the tanner graph of a QC-LDPC Code Based on Protograph Analysis

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PROBLEMS OF INFORMATION TRANSMISSION 2020年第2期56卷 173-184页

作者： Kharin, A., V Zavertkin, K. N. Ovinnikov, A. A. Utkin Ryazan State Radio Engn Univ Fac Radio Engn & Telecommun Dept Theoret Fdn Elect Engn Ryazan Russia

For cycles of length 8 in a tanner graph, we propose an identification procedure based on the analysis of paths in a protograph. We formulate and prove a number of theorems that introduce identification rules for cycles and restrict the number of subgraphs to be analyzed. To distinguish between them, we propose a number of parameters that uniquely determine the group of analyzed paths in the protograph.

关键词： tanner graph protograph expanded graph union of cycles base equation extrinsic message degree LDPC code

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Detecting Cycles of Length 10 in the tanner graph of a QC-LDPC Code Based on Protograph Analysis

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PROBLEMS OF INFORMATION TRANSMISSION 2020年第4期56卷 317-331页

作者： Kharin, A. V. Zavertkin, K. N. Ovinnikov, A. A. Utkin Ryazan State Radio Engn Univ Fac Radio Engn & Telecommun Dept Theoret Fdn Elect Engn Ryazan Russia

We complete the description of the procedure of topological expansion of a bipartite graph without parallel branches on the plane of changing the structure of cycles of length up to 10 inclusive. Based on previous papers, we have extended a set of theorems specifying transformation rules for cycles and paths when passing from a protograph to the tanner graph. We propose a procedure for detecting the existence of a cycle of length 10 in the expanded graph by analyzing the protograph.

关键词： tanner graph protograph expanded graph union of cycles base equation extrinsic message degree LDPC code

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On the performance of tanner graph based and Viterbi Decoding for Erasure Recovery 82

On the performance of Tanner Graph based and Viterbi Decodin...

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IEEE 82nd Vehicular Technology Conference (VTC Fall)

作者： Azeem, Muhammad Moazam Khan, Abdul Baqi Azeem, Uzma Univ Paris 06 Univ Sorbonne UMR 7606 LIP6 F-75005 Paris France Jubail Univ Coll Jubail Ind City 31961 Saudi Arabia Yanbu Univ Coll Yanbu Ind City 31387 Saudi Arabia

ISBN: (纸本)9781479980918

We address the utilization of short erasure correcting codes to recover the erased data in opportunistic spectrum access (OSA) due to collisions among multiple users. The main application of using short codes over long codes is to avoid long delays in the network. Various erasure correcting codes are envisioned and their decoding methods are compared to analyze the erasure recovery capability. The decoding performance of tanner graph based approach is compared with the Exhaustive search based approach that achieves the maximum achievable performance. With the use of extended H matrix there is a huge gain in erasure recovery of lost data and this performance is comparable with Exhaustive search. The Viterbi decoding for erasure recovery is also described with examples and the three techniques are compared to analyze the decoding complexity.

关键词： Oppostunistic Spectrum Access tanner graph Exhaustive search Viterbi Decoding Erasure correcting codes

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A Unifying Framework to Construct QC-LDPC tanner graphs of Desired Girth

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第9期68卷 5802-5822页

作者： Smarandache, Roxana Mitchell, David G. M. Univ Notre Dame Dept Math & Elect Engn Notre Dame IN 46556 USA New Mexico State Univ Klipsch Sch Elect & Comp Engn Las Cruces NM 88003 USA

This paper presents a unifying framework to construct low-density parity-check (LDPC) codes with associated tanner graphs of desired girth. Towards this goal, we highlight the role that a certain square matrix that appears in the product of the parity-check matrix with its transpose has in the construction of codes with graphs of desired girth and further explore it in order to generate the set of necessary and sufficient conditions for a tanner graph to have a given girth between 6 and 12. For each such girth, we present algorithms to construct codes of the desired girth and we show how to use them to compute the minimum necessary value of the lifting factor. For girth larger than 12, we show how to use multi-step graph lifting methods to deterministically modify codes in order to increase their girth. We also give a new perspective on LDPC protograph-based parity-check matrices by viewing them as rows of a parity-check matrix equal to the sum of certain permutation matrices and obtain an important connection between all protographs and those with variable nodes of degree 2. We also show that the results and methodology that we develop for the all-one protograph can be used and adapted to analyze the girth of the tanner graph of any parity-check matrix and demonstrate how this can be done using a well-known irregular, multi-edge protograph specified by the NASA Consultative Committee for Space Data Systems (CCSDS). Throughout the paper, we exemplify our theoretical results with constructions of LDPC codes with tanner graphs of any girth between 6 and 14 and give sufficient conditions for a multi-step lifted parity-check matrix to have girth between 14 and 22.

关键词： Parity check codes Standards NASA Upper bound Terminology Data systems Task analysis Low-density parity-check (LDPC) codes quasi-cyclic codes girth protograph tanner graph

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Doped low-density parity-check codes

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Digital Communications and Networks 2024年第1期10卷 217-226页

作者： Yong Li Rui Liu Xianlong Jiao Youqiang Hu Zhen Luo Francis C.M.Lau College of Computer Science Chongqing UniversityChongqing400044China School of Automation Chongqing UniversityChongqing400044China College of Electronic and Information Engineering Southwest UniversityChongqing400715China Future Wireless Networks and IoT Focusing Area Department of Electronic and Information EngineeringHong Kong Polytechnic UniversityHong KongChina

In this paper,we propose a doping approach to lower the error floor of Low-Density Parity-Check(LDPC)*** doping component is a short block code in which the information bits are selected from the coded bits of the dominant trapping sets of the LDPC ***,an algorithm for selecting the information bits of the short code is proposed,and a specific two-stage decoding algorithm is *** results demonstrate that the proposed doped LDPC code achieves up to 2.0 dB gain compared with the original LDPC code at a frame error rate of 10^(-6)Furthermore,the proposed design can lower the error floor of original LDPC Codes.

关键词： LDPC codes Doped LDPC codes tanner graph Quadratic residue codes Trapping sets

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Construction of Protograph-Based LDPC Codes With Chordless Short Cycles

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IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第1期70卷 51-74页

作者： Amirzade, Farzane Sadeghi, Mohammad-Reza Panario, Daniel Amirkabir Univ Technol Dept Math & Comp Sci Tehran Polytech Tehran *** Iran Carleton Univ Sch Math & Stat Ottawa ON K1S 5B6 Canada

There is a concept in graph theory known as a chord which has not been considered before in relation to trapping sets of tanner graphs. A chord of a cycle is an edge outside the cycle which connects two vertices of that cycle. It is proved that short cycles with a chord are the root of several trapping sets and eliminating them increases the minimum distance dmin of a code. We provide new analytic lower bounds on dmin of LDPC codes with girths 6 and 8 and column weight-y in which the short cycles are all chordless. We prove, analytically, that d(min) >= 2 gamma for girth 6 and d(min )>= 3(gamma-1)(2)/gamma ln gamma-gamma+1 for girth 8. Comparing these bounds with the existing bound gamma + 1 for girth-6 LDPC codes shows the positive and significant influence of eliminating these cycles. A method to construct protograph-based LDPC codes with different girths and free of short cycles with a chord is given which is applicable to any type of protographs, simple and multi-edge, regular and irregular. The conditions to remove small trapping sets from the tanner graph of a multi-edge QC-LDPC code are given. Numerical results indicate that the application of our method to QC-LDPC codes improves existing results.

关键词： LDPC codes girth tanner graph elementary trapping set chordless cycles minimum distance compact QC-LDPC code array-based LDPC code multi-edge QC-LDPC code

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Girth Analysis of Quantum Quasi-Cyclic LDPC Codes

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PROBLEMS OF INFORMATION TRANSMISSION 2024年第2期60卷 71-89页

作者： Amirzade, Farzane Panario, Daniel Sadeghi, Mohammad-Reza Carleton Univ Sch Math & Stat Ottawa ON Canada Amirkabir Univ Technol Dept Math & Comp Sci Tehran Polytech Tehran Iran

Quantum quasi-cyclic LDPC (QQC LDPC) codes, as CSS (Calderbank, Shor, and Steane) codes, are attracting attention because of their good structure and popular channel coding schemes. Fully connected quasi-cyclic LDPC (QC-LDPC) codes with different girths which result in QQC-LDPC codes are investigated. We analytically prove that QC-LDPC codes with column weight at least 3, which yield a QQC-LDPC code, have girth at most 6. To obtain a QQC-LDPC code from QC-LDPC codes with girth more than 6 we should focus on QC-LDPC codes with column weight 2. We present an efficient and practical method to construct these codes with girth at least 8. Then, we extend our method to construct codes with column weight 2 and girth 12, thus reaching the largest possible girth.

关键词： quantum QC-LDPC codes girth tanner graph

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