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Multiple-Insertion-Correcting non-binary Quantum codes and Decoding Algorithm

IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2025年第2期E108A卷 123-128页

作者： Nakamura, Ken Nozaki, Takayuki Yamaguchi Univ Dept Informat Div Fundamental Sci Grad Sch Sci & Technol Innovat Yamaguchi 7538512 Japan Yamaguchi Univ Res Inst Time Studies Yamaguchishi 7538511 Japan

This paper investigates non-binary quantum codes correcting multiple insertion errors. This paper provides an insertion-correcting procedure of the deletion-correcting non-binary codes constructed by Matsumoto and Hag... 详细信息

关键词： decoding algorithm insertion error non-binary codes quantum codes

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SOGRAND Decoding of non-binary Product codes 59

SOGRAND Decoding of Non-binary Product Codes

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59th Annual Conference on Information Sciences and Systems, CISS 2025

作者： Rapp, Lukas Medard, Muriel Duffy, Ken R. Massachusetts Institute of Technology Research Laboratory for Electronics Cambridge United States Northeastern University Dept. of Ece Dept. Mathematics Boston United States

ISBN: (纸本)9798331513269

non-binary product codes are matrices with entries in an extension field, where the rows and columns are protected by linear non-binary component codes. As with binary product codes, non-binary product codes can be decoded via turbo product decoding and are referred to as turbo product codes (TPCs). This paper assesses the performance of non-binary TPCs with the recently developed soft-input soft-output GRAND decoder (SOGRAND). Owing to the more accurate soft output, decoding of non-binary TPCs with SOGRAND outperforms existing turbo product decoders. We demonstrate that non-binary TPCs can also provide a better block error rate performance than a selection of low-density parity-check (LDPC) codes found in the 5G New Radio standard. This extends the design space of TPCs beyond binary TPCs, demonstrating their potential as a viable alternative to LDPC codes. Empirical results show that non-binary TPCs converge in fewer iterations than binary TPCs of the same size. We show TPCs that converge with up to three times fewer decoding iterations than the corresponding LDPC code from the 5G New Radio standard. This indicates they hold promise for low-latency, high-throughput applications. © 2025 IEEE.

关键词： GRAND LDPC non-binary codes product codes

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Adaptive Multiset Stochastic Decoding of non-binary LDPC codes

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2013年第16期61卷 4100-4113页

作者： Ciobanu, Alexandru Hemati, Saied Gross, Warren J. McGill Univ Dept Elect & Comp Engn Montreal PQ H3A 2A7 Canada Linkoping Univ Dept Elect Engn SE-58183 Linkoping Sweden

We propose a non-binary stochastic decoding algorithm for low-density parity-check (LDPC) codes over GF(q) with degree two variable nodes, called Adaptive Multiset Stochastic Algorithm (AMSA). The algorithm uses multisets, an extension of sets that allows multiple occurrences of an element, to represent probability mass functions that simplifies the structure of the variable nodes. The run-time complexity of one decoding cycle using AMSA is O(q) for conventional memory architectures, and O(1) if a custom memory architecture is used. Two fully-parallel AMSA decoders are implemented on FPGA for two (192,96) (2,4)-regular codes over GF(64) and GF(256), both achieving a maximum clock frequency of 108 MHz. The GF(64) decoder has a coded throughput of 65 Mb/s at E-b/N-0 = 2.4 dB when using conventional memory, while a decoder using the custom memory version can achieve 698 Mb/s at the same E-b/N-0. At a frame error rate (FER) of 2 x 10(-6) the GF(64) version of the algorithm is only 0.04 dB away from the floating-point SPA performance, and for the GF(256) code the difference is 0.2 dB. To the best of our knowledge, this is the first fully parallel non-binary LDPC decoder over GF(256) reported in the literature.

关键词： Iterative decoding low-density parity-check code non-binary codes parallel architectures stochastic decoding

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ADMM LP Decoding of non-binary LDPC codes in F_2m

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IEEE TRANSACTIONS ON INFORMATION THEORY 2016年第6期62卷 2985-3010页

作者： Liu, Xishuo Draper, Stark C. Univ Wisconsin Dept Elect & Comp Engn 1415 Johnson Dr Madison WI 53706 USA Univ Toronto Dept Elect & Comp Engn Toronto ON M5S 3G4 Canada

In this paper, we develop efficient decoders for non-binary low-density parity-check codes using the alternating direction method of multipliers (ADMM). We apply ADMM to two decoding problems. The first problem is linear programming (LP) decoding. In order to develop an efficient algorithm, we focus on non-binary codes in fields of characteristic two. This allows us to transform each constraint in F-2m to a set of constraints in F-2 that has a factor graph representation. Applying ADMM to the LP decoding problem results in two types of non-trivial sub-routines. The first type requires us to solve an unconstrained quadratic program. We solve this problem efficiently by leveraging new results obtained from studying the above factor graphs. The second type requires Euclidean projections onto polytopes that are studied in the literature. Such projections can be solved efficiently using off-the-shelf techniques, which scale linearly in the dimension of the vector to project. ADMM LP decoding scales linearly with block length, linearly with check degree, and quadratically with field size. The second problem we consider is a penalized LP decoding problem. This problem is obtained by incorporating a penalty term into the LP decoding objective. The purpose of the penalty term is to make non-integer solutions (pseudocodewords) more expensive and hence to improve decoding performance. The ADMM algorithm for the penalized LP problem requires Euclidean projection onto a polytope formed by embedding the constraints specified by the non-binary single parity-check code, which can be solved by applying the ADMM technique to the resulting quadratic program. Empirically, this decoder achieves a much reduced error rate than LP decoding at low signal-to-noise ratios.

关键词： Error correction coding convex optimization geometry iterative decoding alternating direction method of multipliers non-binary codes

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non-binary LDPC codes for Magnetic Recording Channels: Error Floor Analysis and Optimized Code Design

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IEEE TRANSACTIONS ON COMMUNICATIONS 2016年第8期64卷 3194-3207页

作者： Hareedy, Ahmed Amiri, Behzad Galbraith, Rick Dolecek, Lara Univ Calif Los Angeles Dept Elect Engn Los Angeles CA 90095 USA Hitachi Global Storage Technol Rochester MN 55901 USA

In this paper, we provide a comprehensive analysis of the error floor along with code optimization guidelines for structured and regular non-binary low-density parity-check (NB-LDPC) codes in magnetic recording (MR) applications. While the topic of the error floor performance of binary LDPC codes over additive white Gaussian noise (AWGN) channels has recently received considerable attention, very little is known about the error floor performance of NB-LDPC codes over other types of channels, despite the early results demonstrating superior characteristics of NB-LDPC codes relative to their binary counterparts. We first show that, due to the outer looping between the detector and the decoder in the receiver, the error profile of NB-LDPC codes over partial-response (PR) channels is qualitatively different from the error profile over AWGN channels-this observation motivates us to introduce new combinatorial objects aimed at capturing decoding errors that dominate the PR channel error floor region. We call these objects balanced absorbing sets (BASs), which are viewed as a special subclass of previously introduced absorbing sets (ASs). Aided by these new objects (BASs), we develop a method that combines analytical equations and biased simulations to predict the error floor performance of NB-LDPC codes over PR channels without the need to execute extensive Monte Carlo (MC) simulations. We show that explicitly incorporating the inter-symbol interference of MR channels into our prediction method makes the accuracy of the error floor estimate within 0.2 of an order of magnitude from the traditional MC simulation. In addition, we prove that, due to the more restrictive definition of BASs (relative to the more general class of ASs), an additional degree of freedom can be exploited in the code design for PR channels. We then demonstrate that the proposed code optimization aimed at removing dominant BASs offers performance improvements in the frame error rate in the error floor

关键词： LDPC codes non-binary codes absorbing sets magnetic recording partial-response channels error floor performance error floor prediction

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Bounds on the Belief Propagation Threshold of non-binary LDPC codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2016年第5期62卷 2639-2657页

作者： Geller, Leonid Burshtein, David Tel Aviv Univ Sch Elect Engn IL-69978 Tel Aviv Israel

We consider low-density parity-check (LDPC) code ensembles over non-binary Galois fields when used for transmission over arbitrary discrete memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter, we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.

关键词： Low-density parity-check (LDPC) codes non-binary codes belief propagation decoding

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A Simplified Min-Sum Decoding Algorithm for non-binary LDPC codes

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IEEE TRANSACTIONS ON COMMUNICATIONS 2013年第1期61卷 24-32页

作者： Wang, Chung-Li Chen, Xiaoheng Li, Zongwang Yang, Shaohua LSI Corp Milpitas CA 95035 USA Microsoft Corp Redmond WA 98052 USA

non-binary low-density parity-check codes are robust to various channel impairments. However, based on the existing decoding algorithms, the decoder implementations are expensive because of their excessive computational complexity and memory usage. Based on the combinatorial optimization, we present an approximation method for the check node processing. The simulation results demonstrate that our scheme has small performance loss over the additive white Gaussian noise channel and independent Rayleigh fading channel. Furthermore, the proposed reduced-complexity realization provides significant savings on hardware, so it yields a good performance-complexity tradeoff and can be efficiently implemented.

关键词： Low-density parity-check (LDPC) codes non-binary codes iterative decoding extended min-sum algorithm

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non-binary Turbo codes on Additive Impulsive Noise Channels 10

Non-Binary Turbo Codes on Additive Impulsive Noise Channels

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10th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP)

作者： Abd-Alaziz, Wael Johnston, Martin Le Goff, Stephane Newcastle Univ Sch Elect & Elect Engn Newcastle Upon Tyne Tyne & Wear England

ISBN: (纸本)9781509025268

It is well known that binary error-correcting codes with iterative decoders can achieve near Shannon-limit performance on the additive white Gaussian noise (AWGN) channel, but their performance on more realistic wireless channels can become degraded due to the presence of burst errors or impulsive noise due to interference. A better performing coding scheme is the class of non-binary codes, which are known to be more effective in correcting burst errors, but interestingly there is no research reported in the literature investigating non-binary codes on impulsive noise channels. In this paper, we investigate the performance of non-binary turbo codes defined in a finite field GF(4) on symmetric alpha-stable impulsive noise channels and compare with comparable binary turbo codes. A Cauchy receiver is also employed to mitigate the effects of the channel to assist the turbo decoding. Our simulation results show that although the non-binary turbo code performs similarly to the binary turbo code on the AWGN channel, it achieves a significant coding gain over the binary turbo code as impulsiveness increases.

关键词： non-binary codes Max-Log-Map Turbo codes Impulsive noise Symmetric alpha-stable distributions

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Bounds on the Belief Propagation Threshold of non-binary LDPC codes

Bounds on the Belief Propagation Threshold of Non-Binary LDP...

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IEEE Information Theory Workshop (ITW)

作者： Geller, Leonid Burshtein, David Tel Aviv Univ Sch Elect Engn IL-69978 Tel Aviv Israel

ISBN: (纸本)9781467302234

We consider LDPC code ensembles over non-binary Galois fields, when used for transmission over arbitrary memoryless channels. Belief propagation decoding for these codes has been shown to achieve excellent results. However, computing the decoding threshold using density evolution is usually impractical, since one needs to propagate multi-dimensional probability distributions, and Monte Carlo simulations are required instead. By considering the evolution of the message Bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm. This bound applies for both regular and irregular non-binary LDPC ensembles.

关键词： Low-density parity-check (LDPC) codes non-binary codes belief propagation decoding

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BOUNDS ON THE THRESHOLDS OF non-binary LDPC codes

BOUNDS ON THE THRESHOLDS OF NON-BINARY LDPC CODES

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AIAA 47th Aerospace Sciences Meeting and Exhibit

作者： Levi, Oron Burshtein, David Tel Aviv Univ Sch Elect Engn IL-69978 Tel Aviv Israel

ISBN: (纸本)9781424424818

LDPC codes over non-binary Galois fields were proposed for reliable transmission over arbitrary discrete memoryless channels and in particular for transmission in the bandwidth efficient regime. It was shown that some properties, originally derived for binary LDPC codes, extend to the appropriately defined non-binary ensemble. However, the extension of density evolution to this ensemble results in a prohibitive computational complexity. In this paper we analyze the performance by tracking the evolution of the Bhattacharya noise parameter associated with the decoding messages, and derive upper and lower bounds on its evolution. These bounds are then used to obtain upper and lower bounds on the asymptotic threshold required for reliable communication.

关键词： Low-density parity-check (LDPC) codes non-binary codes iterative decoding

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