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About 1,277 papers for “parity check matrix”

Codes with a Circulant Parity Check Matrix

S.D. Servetto, Richard E. Blahut · 1998

In this work we study codes characterized by the property that their parity check matrix is circulant, i.e., that rows are obtained as all the distinct cyclic shifts of the first row. For these codes, we give simple expressions for their dimension and for a lower bound on their minimum distance. We also present an ${\tt O}(nd)$ algorithm to solve the problem of deciding whether codes satisfying given constraints on their block length $n$, dimension $k$ and minimum distance $d$ exist in this class. Some long binary codes found using this algorithm are compared against the Varshamov-Gilbert bound.

Cited by 0

Tiling parity-check matrix for reduced complexity high throughput low-density parity-check decoders

A. Selvarathinam, Gwan Choi · 2006

An approach for reducing hardware complexity of LDPC decoders is presented in this paper. Low-density parity-check (LDPC) codes have a sparse parity-check matrix (H matrix). In LDPC decoder, the H matrix is stored in memory and contains information about the parity check constraints. The approach presented in this paper constructs several sub-matrices (pseudo random patterns) that are repeatedly used to form the H matrix. The merits of this approach on the decoder architecture are two-fold. First, the switch logic associated with data forwarding in and out of the memory blocks, or alternately the routing of bit nodes to check nodes is simplified. Second, this approach reduces information stored in the design about the H matrix. Thus, the hardware complexity of the decoder is significantly reduced with an added advantage of increased throughput. LDPC code performance simulation results show that the proposed approach does not compromise the bit error rate performance (BER) compared to that of ideal/optimal H matrix for same code length (N = 2040) and rate.

Cited by 1

The Role of Eigenvalues of Parity Check Matrix in Low-Density Parity Check Codes

Debashis Adhikari · 2020

The new developments in coding theory research have revolutionized the application of coding to practical systems. Low-Density Parity Check (LDPC) codes form a class of Shannon limit approaching codes opted for digital communication systems that require high reliability. This thesis investigates the underlying relationship between the spectral properties of the parity check matrix and LDPC decoding convergence. The bit error rate of an LDPC code is plotted for the parity check matrix that has different Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. It is found that for a given (n,k) LDPC code, large SSEM has better error floor performance than low SSEM. The value of SSEM decreases as the sparseness in a parity-check matrix is increased. It was also found from the simulation that long LDPC codes have better error floor performance than short codes. This thesis outlines an approach to analyze LDPC decoding based on the eigenvalue analysis of the corresponding parity check matrix.

Cited by 0Open access

BCK-codes Based on a Parity Check Matrix

Nazanin Keshavarzian, Arsham Borumand Saeid, Abolfazl Tehranian · Fundamenta Informaticae · 2020

Hamming codes are of primary concern in information theory and its applications. Despite a number of researches that have been conducted on such codes and their characterizations, dealing with the properties of previously introduced codes in a BCK-algebraic framework has not been considered in earlier works. This paper investigates a code constructed based on BCK-algebraic models and proposes an algorithm corresponding to the presented code. It is noticeable that the suggested rendered algorithm is also established on the basis of the elements of a BCK-algebra. In fact, both the Hamming distance and dimension, associated with the presented code, can be estimated through a BCK-algebra structure due to the mechanism of its algorithm which is heavily dependent on the parity check matrix. In addition, the way in which the codes are designed contributes substantially to classification of them and to extract greater number of their attributes compared to the previous works. The highlight of the proposed method is that the number of atoms of the BCK-algebra plays a key role in calculation of the Hamming distance and dimension of these codes. Moreover, the obtained codes possess specified and recognizable Hamming distance which are essential in performing error-correcting, error-detecting and decoding tasks.

Cited by 0

Multi-modal Adaptive Steganography Algorithm with Variable Parity Check Matrix

Haitao Song, Guangming Tang, Yifeng Sun et al. · 2018

At present, steganography has a wide range of problems such that the user has lower participation in the steganography process and it's difficult to ensure the security of the steganography algorithm. This is the main cause that steganography cannot go out of laboratory to become practical. In order to solve this problem, this paper proposed a multi-modal adaptive steganography algorithm with variable parity check matrix. Based on the guarantee of high anti-statistical detection, the algorithm gets multiple embedding modification modes by controlling the parity check matrix structure using shared key and establishing the mapping relationship between the key and the height, width of the parity check matrix. Thus, the steganography algorithm becomes secure and controllable for users. The security analysis and experimental results show that the algorithm has higher complexity and can maintain high anti-statistical detection.

Cited by 0

Parity-Check Matrix Calculation for Paraunitary Oversampled DFT Filter Banks

Tanja Karp, Michel Kieffer, Pierre Duhamel · IEEE Transactions on Signal Processing · 2008

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Joint source-channel coding schemes based on oversampled filter banks were recently proposed. Oversampled filter banks are considered as error-correcting codes, extending the notion of code subspace, parity-check matrix, and syndrome to the domain of real and complex numbers. We here present an efficient calculation and implementation of the parity-check polynomial matrices for oversampled discrete Fourier transform (DFT) filter banks. If desired, the calculation of the parity-check polynomials can be performed as part of the prototype filter design procedure. We compare our method to those previously presented in the literature. </para>

Cited by 8

Optimizing the Parity-Check Matrix for Syndrome-Based Neural Decoders

Gastón De Boni Rovella, Meryem Benammar, Tarik Benaddi et al. · 2025

In this work, we investigate the design of the parity-check matrix in the recently introduced Syndrome-Based Neural Decoders, which are powerful neural-network-based channel decoders. We show that the structure of the parity-check matrix has a crucial effect on the decoder’s performance, propose an information-theoretic metric to describe this effect, and suggest an algorithm to construct a parity-check matrix accordingly. The results are illustrated both analytically and numerically, showing a considerable performance gain for the proposed design without any increase in training or decoding complexity.

Cited by 0

Quantum LDPC code orthogonal parity-check matrix pairs H_Gamma, H_Delta

Komoto, Daiki, Kasai, Kenta · DRYAD · 2024

Quantum low-density parity-check (QLDPC) codes require orthogonal parity-check matrices to satisfy the quantum orthogonality condition, ensuring their validity for quantum error correction. This dataset provides orthogonal parity-check matrix pairs $H_\Gamma, H_\Delta$, which are essential for constructing QLDPC codes. The matrices were generated using predefined parameters such as code size, finite field order, and cycle length constraints. The dataset includes a variety of matrix instances stored in structured text format. The dataset was developed as part of a study addressing two fundamental challenges in quantum error correction: approaching the hashing bound—a fundamental performance benchmark and lower bound of quantum capacity—and avoiding the error floor phenomenon, where the error rate stagnates under low noise levels. These challenges have hindered the realization of practical quantum error correction schemes. In our research, we propose a novel construction and decoding method for QLDPC codes that significantly improves upon existing quantum error-correcting codes. Our approach is the first to demonstrate performance approaching the hashing bound while also showing no observable error floor phenomenon in the tested parameter range. This dataset contains the parity-check matrices used in our study, allowing for reproducibility and further exploration of QLDPC code performance. Each parity-check matrix is represented by three files: a structure file containing nonzero element positions, a size descriptor file specifying matrix dimensions and finite field order, and a value file containing corresponding finite field elements. File names follow a systematic convention encoding key parameters such as the number of rows and columns, the field order, and random seed values used for matrix generation. This dataset is a valuable resource for researchers in quantum error correction, enabling exploration of QLDPC codes under different parameter settings. It facilitates testing of decoding algorithms, performance evaluation, and further development of quantum coding theory. The dataset is publicly available on Dryad and can be freely accessed for research and development purposes. There are no legal or ethical constraints associated with the use of this dataset. The matrices were generated entirely within our research and do not contain any personally identifiable or sensitive information. This dataset supports the reproducibility of QLDPC code studies and contributes to advancements in quantum error correction techniques.

Cited by 0Open access

Parity-Check Aided Polarizing Matrix Extension for Efficient IR-HARQ Transmission

Jingqiu Gao, Pingzhi Fan · IEEE Transactions on Communications · 2025

In this paper, by introducing nested parity-check (PC) functions, a new polarizing matrix extension-based hybrid automatic repeat request (PME-HARQ) scheme, called PC-PME-HARQ, is proposed. Such nested PC functions are constructed through a fixed-length cyclic shift register. This approach significantly enhances error correction capability, particularly for short and medium-length packet transmissions. Besides, a dynamic subchannel mapping and optimization framework is developed, which integrates subchannel reliability and distance spectrum to optimize bit-set configurations for varying transmission conditions. Furthermore, the information set approximation puncturing (ISAP) and reliability score-based puncturing (RSP) algorithms have been modified and optimized to outperform the traditional quasi-uniform puncturing (QUP) algorithm, providing improved rate compatibility and reducing Block Error Rate (BLER) in PME-HARQ systems. Simulation results demonstrate that the proposed PC-PME-HARQ scheme achieves substantial performance gains, offering both coding and diversity gains across multiple retransmissions, effectively surpassing the baseline schemes.

Cited by 0

An Authentication Scheme Based on an Low-Density Parity-Check Matrix

2008

Abstract — In this paper, an authentication scheme based on an LDPC (Low-Density Parity-Check) matrix is presented. Upper bounds on probabilities of impersonate attack and substitution attack have been derived using combinatorial argument on an LDPC ensemble. I.

Cited by 0

Ldpc Code Construction using Randomly Permutated Copies of Parity Check Matrix

Amr lulu, Ammar M. Abu‐Hudrouss · An-Najah University Journal for Research - A (Natural Sciences) · 2018

A construction technique is proposed for low-density parity check (LDPC) codes. It uses a base parity check matrix designed from a ran-dom or constructed construction method as Gallager or Quasi-Cyclic LDPC (QC-LDPC) codes in sequence to get codes with multiple lengths and same rate of the base matrix. This is done by using a seed matrix with row and column weights of one, distributed randomly and can be addressed by a number in the base matrix, this method reduces the mem-ory usage and reduce the probability of failing to construct a parity ma-trix by random approaches. Also, by the use of circulant identity seed matrices, QC-LDPC codes can be obtained keeping the same perfor-mance. يتطرق هذا البحث إلى دراسة وعرض طرق بناء الأكواد ذات مصفوفات الفحص قليلة الكثافة بطرق مختلفة مثل الاكواد المبنية بطريقه عشوائية التركيب (مثل أكواد جالاجر)، والاكواد المبنية بطريقه منظمة ذات نسق ونموذج معروف (مثل الأكواد ذات الطبيعة الدائرية). ذلك بالإضافة أن هذه الورقة تقدم طريقة جديدة لبناء الأكواد ذات مصفوفات الفحص قليلة الكثافة والتي تستخدم مصفوفات سابقة البناء كأساس للبناء وللحصول على أكواد ذات أطوال مضاعفة من نفس طول الكود الأساسي وبنفس المعدل. يتم ذلك عن طريق استخدام مصفوفة وحدة موزعة بشكل عشوائي, ويمكن تمييزها برقم عند إدراجها في المصفوفه الأساسية. هذا الأسلوب يقلل من الذاكرة المستخدمة ويقلل من احتمال حدوث فشل في البناء بالطريقة العشوائية. نتائج المحاكاة ايضاً تدل على أن البناء المقترح يزيد من متوسط طول الدوائر في رسومات تانر الدالة على المصفوفة، ويقلل من عدد الدورات ذات الطول الرباعي في المصفوفة الاساسية.

Cited by 1Open access