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University Microfilms International eBooks

Bounds and Constructions for Codes Protecting Against Asymmetric Errors

J. Martin Borden

Published Jan 1, 1981• 0 citations
OpenAlex

Abstract

We study binary error-correcting and error-detecting codes suitable for use on a completely asymmetric or Z channel. This channel accepts inputs of 0 and 1 and reproduces these faithfully as outputs, except that occasionally the input of a 1 will erroneously result in the output of a 0. We construct block codes protecting against this error and bound the performance of the best possible codes.

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Necessary and sufficient conditions on block codes correcting/detecting errors of various types

Jos H. Weber, Cornelis de Vroedt, Dick E. Boekee

Necessary and sufficient conditions are given for block codes to be capable of correcting up to t/sub 1/ symmetric errors, up to t/sub 2/ unidirectional errors, and up to t/sub 3/ asymmetric errors, as well as detecting from t/sub 1/+1 up to d/sub 1/ symmetric errors that are not of the unidirectional type, from t/sub 2/+1 up to d/sub 2/ unidirectional errors that are not of the asymmetric type, and from t/sub 3/+1 up to d/sub 3/ asymmetric errors. Many known conditions on block codes concerning error correction and/or detection appear as special cases of this general result. Further, some codes turn out to have stronger error correcting/detection capabilities than they were originally designed for.>

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Maximum Likelihood Decoding of Codes on the Asymmetric Z-channel

Susanna Spinsante, Ángela I. Barbero

AbstractThe aim of this paper is to extend some basic concepts related to the MaximumLikelihood decoding of codes on the Z-channel, which is a particular, but very important,example of an asymmetric channel. We study distance properties of linear codes overthe Z-channel, in order to de ne a suitable metric for the implementation of a MaximumLikelihood decoder on the channel. A combinatorial expression for an upper bound on theprobability of incorrect Maximum Likelihood decoding is also provided, and comparisonsare given to evaluate the tightness of the bound, when a Hamming code, a Turbo codeand an LDPC code are decoded on the asymmetric channel. 1 Introduction Errors induced by communications on a noisy channel or storage medium can be reduced to anydesired level by proper encoding of the information, as long as the information rate is less thanthe capacity of the channel. This fact was shown in a landmark paper by Shannon, in 1948 [1];since that time, much work has been devoted to the problem of devising ecient encoding anddecoding methods for error control in a noisy environment.In a block-coded system, the source output u represents a k-bit message, the encoder outputc represents an n-symbol codeword, the demodulator output r represents the correspondingbinary received n-tuple and the decoder output u^ represents the k-bit estimate of the encodedmessage. The decoder must produce an estimate u^ of the information sequence u based onthe received sequence r. Equivalently, since there is a one-to-one correspondence between theinformation sequence u and the codeword c, the decoder can produce an estimate ^c of thecodeword c. Obviously, u^ = u if and only if ^c = c. The strategy adopted for choosing an

•Jan 1, 2005•3 citations
We study binary error-correcting and error-detecting codes suitable for use on a completely asymmetric or Z channel. This channel accepts inputs of 0 and 1 and reproduces these faithfully as outputs, except that occasionally the input of a 1 will erroneously result in the output of a 0. We construct block codes protecting against this error and bound the performance of the best possible codes.
We study the error detection problem in q-ary asymmetric channels, wherein every input symbol xiis mapped to an output symbol yisatisfying yi≥xi. A general setting is assumed, where the noise vectors are (potentially) restricted in: 1) the amplitude, yi-xi≤a, 2) the Hamming weight, Σi=1n1{yi≠xi}≤ h, and 3) the total weight, Σi=1n(yi-xi)≤t. Optimal codes detecting these types of errors are described for certain sets of parameters a, h, t, both in the standard and in the cyclic (mod q) version of the problem. It is also demonstrated that these codes are optimal in the large alphabet limit for every a, h, t and every block-length n.
Necessary and sufficient conditions are given for block codes to be capable of correcting up to t/sub 1/ symmetric errors, up to t/sub 2/ unidirectional errors, and up to t/sub 3/ asymmetric errors, as well as detecting from t/sub 1/+1 up to d/sub 1/ symmetric errors that are not of the unidirectional type, from t/sub 2/+1 up to d/sub 2/ unidirectional errors that are not of the asymmetric type, and from t/sub 3/+1 up to d/sub 3/ asymmetric errors. Many known conditions on block codes concerning error correction and/or detection appear as special cases of this general result. Further, some codes turn out to have stronger error correcting/detection capabilities than they were originally designed for.>
AbstractThe aim of this paper is to extend some basic concepts related to the MaximumLikelihood decoding of codes on the Z-channel, which is a particular, but very important,example of an asymmetric channel. We study distance properties of linear codes overthe Z-channel, in order to de ne a suitable metric for the implementation of a MaximumLikelihood decoder on the channel. A combinatorial expression for an upper bound on theprobability of incorrect Maximum Likelihood decoding is also provided, and comparisonsare given to evaluate the tightness of the bound, when a Hamming code, a Turbo codeand an LDPC code are decoded on the asymmetric channel. 1 Introduction Errors induced by communications on a noisy channel or storage medium can be reduced to anydesired level by proper encoding of the information, as long as the information rate is less thanthe capacity of the channel. This fact was shown in a landmark paper by Shannon, in 1948 [1];since that time, much work has been devoted to the problem of devising ecient encoding anddecoding methods for error control in a noisy environment.In a block-coded system, the source output u represents a k-bit message, the encoder outputc represents an n-symbol codeword, the demodulator output r represents the correspondingbinary received n-tuple and the decoder output u^ represents the k-bit estimate of the encodedmessage. The decoder must produce an estimate u^ of the information sequence u based onthe received sequence r. Equivalently, since there is a one-to-one correspondence between theinformation sequence u and the codeword c, the decoder can produce an estimate ^c of thecodeword c. Obviously, u^ = u if and only if ^c = c. The strategy adopted for choosing an