Using codes for error correction and detection (Corresp.)

IEEE Transactions on Information Theory, 2010

Systematic q-ary (q > 2) codes capable of correcting all asymmetric errors of maximum magnitude l, where l q 0 2, are given. These codes are shown to be optimal. Further, simple encoding/decoding algorithms are described. The proposed code can be modified to design codes correcting all symmetric errors of maximum magnitude l, where l q02 2. Index Terms-ary codes, asymmetric channels, error control, error correcting codes, limited magnitude error.

IEEE Transactions on Computers, 1989

We present families of binary systematic codes that can correct t random errors and detect more than t unidirectional errors. As in recent papers, we start by encoding the k information symbols into a codeword of an [n', k, 2t + 11 error correcting code. The second step of our construction involves adding more bits to this linear error correcting code in order to obtain the detection capability of all unidirectional errors. Asymmetric error correcting codes turn out to be a powerful tool in our construction. The resulting codes sensibly improve previous results. Asymptotic estimates and decoding algorithms are presented.

Discrete Applied Mathematics, 2008

The performance of a linear tt-error correcting code over a qq-ary symmetric memoryless channel with symbol error probability εε is characterized by the probability that a transmission error will remain undetected. This probability is a function of εε involving the code weight distribution and the weight distribution of the cosets of minimum weight at most tt. When the undetectable error probability is an increasing function of εε, the code is called tt-proper.The paper presents sufficient conditions for tt-properness and a list of codes known to be proper, many of which have been studied by these sufficient conditions. Special attention is paid to error detecting codes of interest in modern communication.

International Journal of Scientific Research in Science, Engineering and Technology

The integrity of received data is a critical consideration in the design of digital communications and storage systems. The technique involved in attaining data reliability while transmission over a wireless channel is to use Channel Coding. These coding methods involve the use of Error Control Codes and there are two basic ways of controlling errors. They are Automatic Repeat Request and Forward Error Correction. This thesis concentrates on Forward error correction that deals with error detection and error correction. There are two types of error control codes: Block Codes and Convolutional Codes. The extent to which the errors are detected is a measure of the success of the code. The main trade-off in the error correction/detection technique lies in the key parameters involved in evaluating a coding system. Various block codes are analyzed using the performance metrics, namely, Improvement ratio and Error Resilience. It is observed that Cyclic Redundancy Check (CRC) codes showed b...

2012 Information Theory and Applications Workshop, 2012

It is shown that, for all prime powers q and all k ≥ 3, if n ≥ (k − 1)q k − 2 q k −q q−1 , then there exists an [n, k; q] code that is proper for error detection.

2012

The Channel coding deals with error control techniques. If the data at the output of a communications system has errors that are too frequent for the desired use, the errors can often be reduced by the use of a number of techniques. Here the paper present a study on a very simple and adaptable decoder modeling algorithms which we generated the code using Matlab software and tested on three channel-coding techniques namely; Hamming code, convolution code, and uncoded technique for their reliability in error detection over another a comparisons were made between the coded and uncoded techniques and between the two coded techniques, our reliable study shows the reliability of the coded techniques over uncoded one and the reliability of convolutional coded over hamming coded technique, the comparisons were made through their error probabilities under similar constraints of power and information rate.

Journal of Computer Science, 2015

The theory and application of (t-EC-AUED) codes was presented and among the methods proposed in literature, the most efficient was chosen and a software was written for encoding and decoding of the codes. Comparison and evaluation of the construction techniques were carried out. A tError Correction (EC)/All Unidirectional Error Detection (AUED) codes are constructed by appending a single check symbol to a linear t-EC code to achieve the AUED property.

Computing Research Repository, 2006

We consider codes over the alphabet Q = {0,1, . . . , q − 1} intended for the control of unidirectional errors of level ℓ. That is, the transmission channel is such that the received word cannot contain both a component larger than the transmitted one and a component smaller than the transmitted one. Moreover, the absolute value of

ArXiv, 2016

We introduce the concept of an \ff-maximal error-detecting block code, for some parameter \ff{} between 0 and 1, in order to formalize the situation where a block code is close to maximal with respect to being error-detecting. Our motivation for this is that constructing a maximal error-detecting code is a computationally hard problem. We present a randomized algorithm that takes as input two positive integers $N,\ell$, a probability value \ff, and a specification of the errors permitted in some application, and generates an error-detecting, or error-correcting, block code having up to $N$ codewords of length $\ell$. If the algorithm finds less than $N$ codewords, then those codewords constitute a code that is \ff-maximal with high probability. The error specification (also called channel) is modelled as a transducer, which allows one to model any rational combination of substitution and synchronization errors. We also present some elements of our implementation of various error-det...

Journal of Electrical Engineering

In [1] a new family of error detection codes called Weighted Sum Codes was proposed. In [2] it was noted, that these codes are equivalent to lengthened Reed Solomon Codes, and shortened versions of lengthened Reed Solomon codes respectively, constructed over GF(2^(h/2)). It was also shown that it is possible to use these codes for correction of one error in each codeword over GF(2^(h/2)). In [3] a class of modified Generalized Weighted Sum Codes for single error and conditionally double error correction were presented. In this paper we present a new family of double error – correcting codes with code distance dm = 5. Weight spectrum for [59,49,5] code constructed over GF(8) which is an example of the new codes was obtained by computer using its dual [4]. The code rate of the new codes are higher than the code rate of ordinary Reed Solomon codes constructed over the same �finite fi�eld.

Computers & Mathematics with Applications, 1990

Al~tract-Codes which can correct t symmetric errors and detect all unidirectional errors have been shown to be useful in fault-tolerant applications. They provide protection against transient, intermittent and permanent faults. Efficient codes have already been developed for this purpose. However, for most of these codes it is not possible to obtain a simple encoder and decoder. We have developed a systematic terror correcting/all unidirectional error detecting code, which is comparable in redundancy to the most efficient known systematic codes with the same capability. In addition, our code is easy to encode and decode.

Designs, Codes and Cryptography, 1997

This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different codewords. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. This is accomplished by a new method called parity sectioned reduction. In this method, the parity check matrix of a uniform error correcting linear code is reduced by dropping some rows and columns and the error range inequalities are modified.

Physical Review Letters, 2000

The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly constructed sparse matrices; the number of nonzero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the Thouless-Anderson-Palmer approach which is identical to the commonly used belief-propagation-based decoding. PACS numbers: 89.90. + n, 02.50.2r, 05.50. + q, 75.10.Hk The ever increasing information transmission in the modern world is based on communicating messages reliably through noisy transmission channels; these can be telephone lines, magnetic storing media, etc. Errorcorrecting codes play an important role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission and decoding the corrupted received codeword for retrieving the original message. In his groundbreaking papers, Shannon [1] analyzed the capacity of communication channels, setting an upper bound to the achievable noise-correction capability of codes, given their code (or symbol) rate. The latter represents the ratio between the number of bits in the original message and in the transmitted codeword.

Lecture Notes in Electrical Engineering, 2021

Interactive Theorem Proving, 2015

By adding redundancy to transmitted data, error-correcting codes (ECCs) make it possible to communicate reliably over noisy channels. Minimizing redundancy and (de)coding time has driven much research, culminating with Low-Density Parity-Check (LDPC) codes. At first sight, ECCs may be considered as a trustful piece of computer systems because classical results are well-understood. But ECCs are also performance-critical so that new hardware calls for new implementations whose testing is always an issue. Moreover, research about ECCs is still flourishing with papers of ever-growing complexity. In order to provide means for implementers to perform verification and for researchers to firmly assess recent advances, we have been developing a formalization of ECCs using the SSReflect extension of the Coq proof-assistant. We report on the formalization of linear ECCs, duly illustrated with a theory about the celebrated Hamming codes and the verification of the sum-product algorithm for decoding LDPC codes.