Asymmetric Binary Channel Research Articles

Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels

In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees.

Efficient LLR Calculation for Non-Binary Modulations over Fading Channels

Log-likelihood ratio (LLR) computation for non-binary modulations over fading channels is complicated. A measure of LLR accuracy on asymmetric binary channels is introduced to facilitate good LLR approximations for non-binary modulations. Considering piecewise linear LLR approximations, we prove convexity of optimizing the coefficients according to this measure. For the optimized approximate LLRs, we report negligible performance losses compared to true LLRs.

Information flow on trees

Consider a tree network $T$, where each edge acts as an independent copy of a given channel $M$, and information is propagated from the root. For which $T$ and $M$ does the configuration obtained at level $n$ of $T$ typically contain significant information on the root variable? This problem arose independently in biology, information theory and statistical physics. For all $b$, we construct a channel for which the variable at the root of the break $b$-ary tree is independent of the configuration at the second level of that tree, yet for sufficiently large $B>b$, the mutual information between the configuration at level $n$ of the $B$-ary tree and the root variable is bounded away from zero for all $n$. This construction is related to Reed--Solomon codes. We improve the upper bounds on information flow for asymmetric binary channels (which correspond to the Ising model with an external field) and for symmetric $q$-ary channels (which correspond to Potts models). Let $\lam_2(M)$ denote the second largest eigenvalue of $M$, in absolute value. A CLT of Kesten and Stigum implies that if $b |\lam_2(M)|^2 >1$, then the census of the variables at any level of the $b$-ary tree, contains significant information on the root variable. We establish a converse: If $b |\lam_2(M)|^2 < 1$, then the census of the variables at level $n$ of the $b$-ary tree is asymptotically independent of the root variable. This contrasts with examples where $b |\lam_2(M)|^2 <1$, yet the configuration at level $n$ is not asymptotically independent of the root variable.

Noise-enhanced capacity via stochastic resonance in an asymmetric binary channel

A nonlinear system is considered where an aperiodic binary input signal is added to an arbitrarily distributed noise and compared to a fixed threshold to determine the binary output signal. Noise enhancement of the transmission of the aperiodic signal via stochastic resonance is demonstrated and studied in this nonlinear information channel. The characterization developed goes up to the calculation of the information capacity of the channel, defined as the maximal achievable input-output transinformation occurring when the statistics of the input signal is matched to the noise. It is then demonstrated that a regime exists where the information capacity of the channel can be increased by means of an increase of the noise, up to an optimal noise level where the capacity resonates at a maximum value. The influence on this resonance of the noise distribution is also studied.

Forward error-correction codes in incoherent optical fibre CDMA networks

The performance of incoherent optical CDMA systems can be improved by using forward error-correction codes. Such a system behaves as an asymmetric binary channel due to optical multiple access interference. To effectively combat the interference, asymmetric error correction (AEC) codes are recommended. The use of AEC binary block codes not only reduces the bit error rate, but also increases the maximum number of users in a constant-bandwidth network more efficiently than symmetric error correction binary block codes.

Optimal error detection codes for completely asymmetric binary channels

The (n/2)-out-of-n code is proved to be the least redundant binary block code which permits the detection of all errors in completely asymmetric channels. It is then proved that the sum code of Berger, Smith, and Freiman is the least redundant of all separable codes of this type. The redundancies of the sum and (n/2)-out-of-n codes are then compared and it is shown that the former is asymptotically twice as redundant as the latter. An efficient method of constructing separable codes which detect up to a given number, but not all, asymmetric errors is included as an appendix.

Single error-correcting codes for asymmetric binary channels

In a highly-asymmetric binary channel it may be necessary to correct only those errors which result from incorrect transmission of one of the two code elements. Minimum weight-distance relationships and rules for generating single-error correcting codes in such situations are given. More code characters are generally obtained for a given character length than are obtained with codes designed for single-error correction in symmetric channels. Examples are given, including one which specifies the code which results in the highest average probability of correct transmission of equiprobable messages through a highly-asymmetric channel.

Multi-error correcting codes for a binary asymmetric channel

In an asymmetric binary channel, it may be sufficient to correct single O-errors and detect double 0-errors, for example, while correcting double 1-errors and detecting quadruple 1-errors. (A double 1-error is said to occur when two of the l's of an input code character are delivered as O's at the output of the channel.) Minimum distance requirements are given for pairs of code characters of a code which corrects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple 1-errors, detects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+a)</tex> -tuple 1-errors, corrects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</tex> -tuple 0-errors, and detects <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(j+b)</tex> -tuple O-errors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k,a,j</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> , are non-negative integers with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k &gt; j</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a \geq b</tex> ). These requirements are weaker than those for a symmetrical <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple error correcting, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+a)</tex> -tuple error detecting, code and hence may be used to generally obtain more code characters for a given character length than are obtainable in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k, (k+a)</tex> -case. If the channel is highly asymmetric, it may be sufficient to detect and correct only one type of error. An earlier paper considered the case of single 1-error correction and showed that it was always possible to obtain more code characters than exist in known single error correcting codes of equivalent character length except in cases where the symmetric code is "close-packed." In this paper codes are developed for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple 1-error correction which also yield more code characters than symmetrical <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -tuple error correcting codes of the same length. The correction scheme is generally symbol-correcting, but may require message-correction of binary sequences whose length is approximately <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k+l)^{-1}</tex> that of the code characters. A double 1-error correcting code is discussed in some detail and examples of code generation and correction are included.