Error-correcting Properties Research Articles

Polyadic codes of prime power length

Polyadic codes constitute a special class of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error-correcting properties. In this paper, we give necessary and sufficient conditions for the existence of polyadic codes of prime power length. Examples of some good codes arising from the family of polyadic codes of prime power length are also given.

A simple baseband transmission scheme for power line channels

We propose a simple pulse-amplitude modulation (PAM)-based coded modulation scheme that overcomes two major constraints of power line channels, viz., severe insertion-loss and impulsive noise. The scheme combines low-density parity-check (LDPC) codes, along with cyclic random-error and burst-error correction codes to achieve high-spectral efficiency, low decoding complexity, and a high degree of immunity to impulse noise. To achieve good performance in the presence of intersymbol interference (ISI) on static or slowly time-varying channels, the proposed coset-coding is employed in conjunction with Tomlinson-Harashima precoding and spectral shaping at the transmitter. In Gaussian noise, the scheme performs within 2 dB of unshaped channel capacity at a bit-error rate (BER) of 10/sup -11/, even with (3,6)-regular LDPC codes of modest length (1000-2000 bits). To mitigate errors due to impulse noise (a combination of synchronous and asynchronous impulses), a multistage interleaver is proposed, each stage tailored to the error-correcting property of each layer of the coset decomposition. In the presence of residual ISI, colored Gaussian noise, as well as severe synchronous and asynchronous impulse noise, the gap to Shannon capacity of the scheme to a Gaussian-noise-only channel is 5.5 dB at a BER of 10/sup -7/.

Weak pairwise correlations imply strongly correlated network states in a neural population

Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher-order interactions among large groups of elements have an important role. Here we show, in the vertebrate retina, that weak correlations between pairs of neurons coexist with strongly collective behaviour in the responses of ten or more neurons. We find that this collective behaviour is described quantitatively by models that capture the observed pairwise correlations but assume no higher-order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behaviour. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.

Encoding Strategy for Maximum Noise Tolerance Bidirectional Associative Memory

In this paper, the basic bidirectional associative memory (BAM) is extended by choosing weights in the correlation matrix, for a given set of training pairs, which result in a maximum noise tolerance set for BAM. We prove that for a given set of training pairs, the maximum noise tolerance set is the largest, in the sense that this optimized BAM will recall the correct training pair if any input pattern is within the maximum noise tolerance set and at least one pattern outside the maximum noise tolerance set by one Hamming distance will not converge to the correct training pair. This maximum tolerance set is the union of the maximum basins of attraction. A standard genetic algorithm (GA) is used to calculate the weights to maximize the objective function which generates a maximum tolerance set for BAM. Computer simulations are presented to illustrate the error correction and fault tolerance properties of the optimized BAM.

Parallel versus sequential updating for belief propagation decoding

A sequential updating scheme (SUS) for the belief propagation algorithm is proposed, and is compared with the parallel (regular) updating scheme (PUS). Simulation results on various codes indicate that the number of iterations of the belief algorithm for the SUS is about one half of the required iterations for the PUS, where both decoding algorithms have the same error correction properties. The complexity per iteration for both schemes is similar, resulting in a lower total complexity for the SUS, furthermore, the SUS utilizes significantly less memory during the decoding process. We demonstrate that the acceleration in convergence time is related to the inter-iteration information sharing, which is a property of only the SUS, and which increases the “correction gain” per iteration. Finally, the connection between the dynamics of error correcting codes and physical systems is discussed.

On design of error-correcting reversible variable length codes

We propose an algorithm for construction of reversible variable length codes (RVLCs) with good error-correcting properties. The error-correcting properties are evaluated by a metric called the free distance, which is always greater than one in the case of the proposed RVLCs. Since variable length codes (VLCs) typically have free distance equal to one, the proposed RVLCs exhibit significant improvement in symbol error rate relative to VLCs constructed using standard methods.

An optimization-based design procedure for asymmetric bidirectional associative memories

In this letter, we consider the problem of designing asymmetric bidirectional associative memories (ABAM). Based on a newly derived theorem for the ABAM model, we propose an optimization-based design procedure for obtaining an ABAM that can store given bipolar vector pairs with certain error correction properties. Our design procedure consists of generalized eigenvalue problems, which can be efficiently solved by recently developed interior point methods. The validity of the proposed method is illustrated by a design example.

Selection of Multi-h modulation sets using a stochastic search algorithm

Multi-h CPFSK modulation is simple to implement, spectrally efficient, and has inherent error correction properties. Multi-h requires large parameter sets to obtain high coding gain. Large numbers of combinations may need to be investigated. A fast efficient and thorough search needs to be performed. A search method based on a genetic algorithm known as ‘population based incremental learning’ (PBIL) is described. A number of sets of modulation indices are shown and their performance is evaluated.

Error correction with frames: the Burt-Laplacian pyramid

The study of multiresolution representations of images led to the development of the pyramid and wavelet representations. The redundancy in the pyramid representation bestows on it the properties of an error-correcting code. The error-correcting properties of the pyramids studied are a property of the representation itself and not a result of making any assumptions about the original images. The article extends the error-reduction properties of frames to an error-correcting routine. It leads to complete noise removal for sparse noise for a class of frames. The results for images using Laplacian pyramids show an additional 10-dB improvement over the previously demonstrated noise reduction for frames.

Modelling of the Bonhoeffer-effect during LTP learning

We investigate the effect of non-local synaptic enhancement within the framework of networks of formal neurons. Enhancement of nearby synapses on a common axon without simultaneous post- and presynaptic stimulation is modelled by a generalized Hebbian rule. The imprinted patterns of activity are no longer attractors of the network dynamics, but are processed instead. The network acts as a low band-pass filter for spatially uncorrelated patterns and can discriminate spatially correlated patterns, according to their degree and range of correlation. While the network is processing information, it still maintains some of the error-correcting properties of attractor neural networks.

Modelling of the Bonhoeffer-effect during LTP learning

We investigate the effect of non-local synaptic enhancement within the framework of networks of formal neurons. Enhancement of nearby synapses on a common axon without simultaneous post- and presynaptic stimulation is modelled by a generalized Hebbian rule. The imprinted patterns of activity are no longer attractors of the network dynamics, but are processed instead. The network acts as a low band-pass filter for spatially uncorrelated patterns and can discriminate spatially correlated patterns, according to their degree and range of correlation. While the network is processing information, it still maintains some of the error-correcting properties of attractor neural networks.

DC-constrained error-correcting codes with small running digital sum

The authors investigate the problem of evaluating the possible size of error-correcting codes with code words taken from a subset of Hamming spaces. This is an example of the problem of constructing codes in irregular subsets of Hamming spaces. The authors examine the theoretical restrictions on the parameters of error-correcting codes in the asymptotic case (semi-finite sequence) when the recursive digital sum is upper-bounded by some small constant. The bounds allow demonstration of the existence of long codes that have good error-correcting properties and that satisfy some restrictions that are natural for optical and magnetic recording.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Binary adaptive nonlinear multiplexing

Error-correcting properties of a binary adaptive nonlinear multiplexing system which trades channel activity for redundancy are studied. The main results are a family of data channel carriers, a way to establish the multiplex errorcorrection power, and the specification of desirable properties for a set of data channel carriers. Channel activity is extracted at the receiver, directly from the multiplexed signal.

Dual Versus Triplication Reliability Estimations

The system mean time to failure (MTTF), a measure of reliability, depends largely on the failure rate, the mean time to repair, and the coverage factor. All contribute to the calculation of MTTF and influence the system design in terms of reliability objectives. In dynamically redundant systems, such as dual systems, coverage plays an extremely important role. It is a probability that the system will recover from a single fault. This involves both hardware and software components to properly detect, isolate, and reconfigure around a faulty module without adversely affecting the system operation. A small reduction in coverage can affect the MTTF dramatically. In a static redundant triple modular redundancy (TMR) system, the inherent error detection and correction property gives a near-perfect coverage for the first detected fault. As a result, the calculated MTTF of a TMR configuration is substantially greater than that of a dual system. This paper discusses and compares the reliability estimation of dual and TMR systems.

Noise immunity of the genetic code

Error detection and correction properties are fundamental for informative codes. Hamming's distance allows us to study this noise resistance. We present codes characterized by the resistance optimization to nonsense mutational effects. The calculation of the cumulated Hamming's distance allowing to determine the number of optimal codes and their structure can be detailed. The principle of these laws of optimization of resistance consists of choosing constituent codons connected by mutational neighbouring in such a way that random application of mutations on such a code minimize the occurrence of nonsense n-uplets or terminators. New coding symmetries are then described and screened using Galois's polynomials properties and Baudot's code. Such a study can be applied to any length of the codons. Here we present the principles of this optimization for the most simple doublet codes. Another constraint is discussed: the distribution of optimal subcodes for synonymity and the frequencies of utilization of the different codons. We compare these results to those of the present genetic code, and we observe that all coded amino acids (except the particular case of SER) are using optimal sub-codes of synonymity. This work suggests that the appearance of the genetic code was provoked by mutations while optimizing on several levels its resistance to their effects. Thus genetic coding would have been the best automata that could be produced in prebiotic conditions.

A class of shortened cyclic codes for a compound channel (Corresp.)

This correspondence presents a class of shortened cyclic codes that are suitable for application on a compound channel. The Chinese remainder theorem is applied to the proof of the error-correcting properties of this class of shortened cyclic codes.

Bandwidth Compaction Codes for Communications

The familiar error-correction codes allow a reduction in the required signal-to-noise ratio at the expense of an increase in bandwidth. Here we reverse the problem and investigate codes that permit a reduced bandwidth at the expense of an increase in the required signal-to-noise ratio. Theoretical properties of these bandwidth compaction codes have been published previously. This paper emphasizes the tradeoff between bandwidth and signal-to-noise ratio when the codes are used. The inherent error detection and error correction properties of the codes are also explored.

Correlation-recovered adaptive majority multiplexing

A novel method of multiplexing based on a pattern-recognition technique is described, in which binary data are combined for transmission over a common medium by modulation onto binary codewords. A majority-logic gate forms the transmitted (binary) signal from the modulated codewords. Demodulation is effected by a correlation process using digital matched filters. The system exhibits an adaptive property in the relationship between the channel error-correcting properties and the number of channels in use. Theoretical and experimental results are reproduced, and a description of a prototype equipment is included.

Cyclic and multiresidue codes for arithmetic operations

In this paper, the cyclic nature of AN codes is defined after a brief summary of previous work in this area is given. New results are shown in the determination of the range for single-error-correcting AN codes when A is the product of two odd primes p_1 and p_2 , given the orders of 2 modulo p_1 and modulo p_2 . The second part of the paper treats a more practical class of arithmetic codes known as separate codes. A generalized separate code, called a multiresidue code, is one in which a number N is represented as \begin{equation} [N, \mid N \mid _ {m1}, \mid N \mid _{m2}, \cdots , \mid N \mid _{mk}] \end{equation} where m_i are pairwise relatively prime integers. For each AN code, where A is composite, a multiresidue code can be derived having error-correction properties analogous to those of the AN code. Under certain natural constraints, multiresidue codes of large distance and large range (i.e., large values of N ) can be implemented. This leads to possible realization of practical single and/or multiple-error-correcting arithmetic units.

Gradient, error-correction identification algorithms

This paper demonstrates a theoretical basis for identifying stationary or slowly varying parameters by means of a gradient algorithm. A priori conditions under which this algorithm is uniformly asymptotically stable (convergent) in the large are given. Special attention is directed at choosing the weighting matrix in the algorithm. The matrix is obtained in two distinct ways. In the first way, the weighting matrix is found as a consequence of an error-correction property. This property is given a rigorous mathematical basis in this paper. In the second way, the weighting matrix is found by maximizing the rate of change of a Lyapunov function. It is shown that the Lyapunov-optimum gain matrix is also an error-correction gain matrix. Applications for the identification algorithm as well as examples that illustrate it are presented. It is shown, by example, that it is better to use a matrix of gains rather than a scalar gain in the algorithm. Extensions to nonstationary identification and identification in the presence of noisy measurements are discussed briefly.