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About 8,600 papers for “parity check”

Parity Check

Encyclopedic Dictionary of Genetics, Genomics and Proteomics · 2004

Cited by 0

Parity-Check Code Decoder

Tong Zhang, Keshab K. Parhi · 2002

Because of their excellent error-correcting performance, low-density parity-check (LDPC) codes have recently attracted a lot of attention. In this paper, we are interested in the practical LDPC code decoder hardware implementations. The direct fully parallel decoder implementation usually incurs too high hardware complexity for many real applications, thus partly parallel decoder design approaches that can achieve appropriate trade-offs between hardware complexity and decoding throughput are highly desirable. Applying a joint code and decoder design methodology, we develop a high-speed (3,k)-regular LDPC code partly parallel decoder architecture based on which we implement a 9216-bit, rate-1/2(3, 6)-regular LDPC code decoder on Xilinx FPGA device. This partly parallel decoder supports a maximum symbol throughput of 54 Mbps and achieves BER 10 −6 at 2 dB over AWGN channel while performing maximum 18 decoding iterations.

Cited by 0

Adding a parity-check bit

Juriaan Simonis · IEEE Transactions on Information Theory · 2000

The article gives a new condition for a q-ary linear code of length n and minimum distance d to be extendable to a code of the same dimension, length n+1, and minimum distance >d.

Cited by 27

Pseudocodewords of Parity-Check Codes

Wittawat Kositwattanarerk · TigerPrints (Clemson University) · 2011

The success of modern algorithms for the decoding problem such as message-passing iterative decoding and linear programming decoding lies in their local nature. This feature allows the algorithms to be extremely fast and capable of correcting more errors than guaranteed by the classical minimum distance of the code. Nonetheless, the performance of these decoders depends crucially on the Tanner graph representation of the code. In order to understand this choice of representation, we need to analyze the pseudocodewords of the Tanner graph of a code. These pseudocodewords are outputs of local decoding algorithms which may not be legitimate codewords. In this dissertation, we introduce a lifted fundamental cone and show that there is a one-to-one correspondence between graph cover pseudocodewords of a binary code and integer points in the lifted fundamental cone. We use this fact to prove the rationality of the generating function of the pseudocodewords for a general binary parity-check code. Our approach also yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Understanding irreducible pseudocodewords is crucial to determining the best representation of a code. Moreover, combining these techniques with the recent characterization of fundamental cone over F_3, we can analyze ternary parity-check codes. Finally, we make progress in the study of more general nonbinary codes by determining constraints satisfied by all pseudocodewords of a code over F_p where p is prime.

Cited by 0Open access

Low density parity check codes

Ortega Sánchez-Colomer, Tomás · QRU Quaderns de Recerca en Urbanisme · 2021

Low Density Parity Check codes, LDPCs for short, are a family of codes which have shown near optimal error-correcting capabilites. They were proposed in 1963 by Robert Gallager in his PhD thesis. While he proved that probabilistic constructions of random LDPCs gave asymptotically good linear codes, they were largely abandoned due to the lack of computing power to make them practically feasable. They enjoyed a re-birth during the coding revolution of the 1980's, and thanks to the developement of expander graph theory, it was proven that they can be encoded and decoded in linear time. This thesis will review the main results through this journey. Nowadays, LDPCs appear in a plethora of commercial applications. The codes used in practice and the techniques that were employed to construct them will also be explored in this work. Finally, a new family of LDPCs will be proposed, which will be constructed from incidence structures called generalized quadrangles, and perform markedly better than random codes.

Cited by 0Open access

Low-Density Parity-Check Codes

Ian F. Blake · Cambridge University Press eBooks · 2024

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 1

Low-Density Parity-Check Codes

Robert G. Gallager · The MIT Press eBooks · 1963

This is a complete presentation of all important theoretical and experimental work done on low-density codes. Low-density coding is one of the three techniques thus far developed for efficient communication over noisy channels with an arbitrarily low probability of error. A principal result of information theory is that if properly coded information is transmitted over a noisy channel at a rate below channel capacity, the probability of error can be made to approach zero exponentially with the code length. Any practical use of this theorem, however, requires a coding scheme in which the cost of storage and computation equipment grows slowly with code length. The present book analyzes a class of coding schemes for which costs grow approximately linearly with code length. It demonstrates that error probability approaches zero exponentially with a root of the block length and cites experimental evidence that this coding scheme has profitable aplicability in many communications situations.

Cited by 4,340

Low-density parity-check codes

Marc Mézard, Andrea Montanari · 2009

Abstract Low-density parity-check (LDPC) codes are among the most efficient error correcting codes in use. This chapter introduces an important family of LDPC ensembles, based on random factor graphs, and studies some of their basic properties. It focuses on performances under optimal decoding, when no constraint is imposed on the computational complexity of the decoding procedure. Bounds in their performances are derived through an analysis of the geometric properties of their codebook. In particular, it shows that appropriately chosen LDPC ensembles allow for communication reliably at rates close to Shannon's capacity.

Cited by 7

On Sparse Parity Check Matrices

Hanno Lefmann, Pavel Pudlák, Petr Savick'y · 1999

We consider the extremal problem to determine the maximal number N(m; k; r) of columns of a 0-1-matrix with m rows and at most r ones in each column such that each k columns are linearly independent modulo 2. For each fixed k 1 and r 1, we shall prove a probabilistic lower bound N(m; k; r) = \\Omega\\Gamma m kr=2(k\\Gamma1) ); for k a power of 2, we prove an upper bound N(m; k; r) = O(m dkr=(k\\Gamma1)e=2 ) which matches the lower bound for infinitely many values of r. We give some explicit constructions. 1 Introduction We shall consider matrices M over the two-element field GF 2 . If each k column vectors of M are linearly independent, we say that the columns are k-wise independent. By a (k; r)-matrix we mean a matrix, where the column vectors are k-wise independent and each column contains at most r ones. We denote by N(m; k; r) the maximal number of columns in a (k; r)-matrix with m rows. The aim of this paper is to give estimates on the growth of the function N(m; k; r). Matric...

Cited by 0