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All sourcesNASA NTRSDOE OSTI
NASA NTRS2009

LDPC Codes with Minimum Distance Proportional to Block Size

Low-density parity-check (LDPC) codes characterized by minimum Hamming distances proportional to block sizes have been demonstrated. Like the codes mentioned in the immediately preceding article, the present codes are error-correcting codes suitable for use in a variety of wireless data-communication systems that include noisy channels. The previously mentioned codes have low decoding thresholds and reasonably low error floors. However, the minimum Hamming distances of those codes do not grow linearly with code-block sizes. Codes that have this minimum-distance property exhibit very low error floors. Examples of such codes include regular LDPC codes with variable degrees of at least 3. Unfortunately, the decoding thresholds of regular LDPC codes are high. Hence, there is a need for LDPC codes characterized by both low decoding thresholds and, in order to obtain acceptably low error floors, minimum Hamming distances that are proportional to code-block sizes. The present codes were developed to satisfy this need. The minimum Hamming distances of the present codes have been shown, through consideration of ensemble-average weight enumerators, to be proportional to code block sizes. As in the cases of irregular ensembles, the properties of these codes are sensitive to the proportion of degree-2 variable nodes. A code having too few such nodes tends to have an iterative decoding threshold that is far from the capacity threshold. A code having too many such nodes tends not to exhibit a minimum distance that is proportional to block size. Results of computational simulations have shown that the decoding thresholds of codes of the present type are lower than those of regular LDPC codes. Included in the simulations were a few examples from a family of codes characterized by rates ranging from low to high and by thresholds that adhere closely to their respective channel capacity thresholds; the simulation results from these examples showed that the codes in question have low error floors as well as low decoding thresholds. As an example, the illustration shows the protograph (which represents the blueprint for overall construction) of one proposed code family for code rates greater than or equal to 1.2. Any size LDPC code can be obtained by copying the protograph structure N times, then permuting the edges. The illustration also provides Field Programmable Gate Array (FPGA) hardware performance simulations for this code family. In addition, the illustration provides minimum signal-to-noise ratios (Eb/No) in decibels (decoding thresholds) to achieve zero error rates as the code block size goes to infinity for various code rates. In comparison with the codes mentioned in the preceding article, these codes have slightly higher decoding thresholds.

Divsalar, Dariush↗
NASA NTRS1989

Associative memory - An optimum binary neuron representation

Convergence mechanism of vectors in the Hopfield's neural network is studied in terms of both weights (i.e., inner products) and Hamming distance. It is shown that Hamming distance should not always be used in determining the convergence of vectors. Instead, weights (which in turn depend on the neuron representation) are found to play a more dominant role in the convergence mechanism. Consequently, a new binary neuron representation for associative memory is proposed. With the new neuron representation, the associative memory responds unambiguously to the partial input in retrieving the stored information.

Awwal, A. A.↗
NASA NTRS1989

Machine parts recognition using a trinary associative memory

The convergence mechanism of vectors in Hopfield's neural network in relation to recognition of partially known patterns is studied in terms of both inner products and Hamming distance. It has been shown that Hamming distance should not always be used in determining the convergence of vectors. Instead, inner product weighting coefficients play a more dominant role in certain data representations for determining the convergence mechanism. A trinary neuron representation for associative memory is found to be more effective for associative recall. Applications of the trinary associative memory to reconstruct machine part images that are partially missing are demonstrated by means of computer simulation as examples of the usefulness of this approach.

Awwal, Abdul Ahad S.↗
NASA NTRS2017

Analog Correlator Based on One Bit Digital Correlator

A two input time domain correlator may perform analog correlation. In order to achieve high throughput rates with reduced or minimal computational overhead, the input data streams may be hard limited through adaptive thresholding to yield two binary bit streams. Correlation may be achieved through the use of a Hamming distance calculation, where the distance between the two bit streams approximates the time delay that separates them. The resulting Hamming distance approximates the correlation time delay with high accuracy.

Prokop, Norman↗
NASA NTRS1989

Auto- and hetero-associative memory using a 2-D optical logic gate

An optical associative memory system suitable for both auto- and hetero-associative recall is demonstrated. This system utilizes Hamming distance as the similarity measure between a binary input and a memory image with the aid of a two-dimensional optical EXCLUSIVE OR (XOR) gate and a parallel electronics comparator module. Based on the Hamming distance measurement, this optical associative memory performs a nearest neighbor search and the result is displayed in the output plane in real-time. This optical associative memory is fast and noniterative and produces no output spurious states as compared with that of the Hopfield neural network model.

Chao, Tien-Hsin↗
NASA NTRS1992

Auto and hetero-associative memory using a 2-D optical logic gate

An optical system for auto-associative and hetero-associative recall utilizing Hamming distance as the similarity measure between a binary input image vector V(sup k) and a binary image vector V(sup m) in a first memory array using an optical Exclusive-OR gate for multiplication of each of a plurality of different binary image vectors in memory by the input image vector. After integrating the light of each product V(sup k) x V(sup m), a shortest Hamming distance detection electronics module determines which product has the lowest light intensity and emits a signal that activates a light emitting diode to illuminate a corresponding image vector in a second memory array for display. That corresponding image vector is identical to the memory image vector V(sup m) in the first memory array for auto-associative recall or related to it, such as by name, for hetero-associative recall.

Chao, Tien-Hsin↗
NASA NTRS1973

Algebraic decoding of block codes over a q-ary input, Q-ary output channel, Q greater than q.

Decoding algorithms designed for one output alphabet are shown to be effectively usable for channels with a different output alphabet. The described technique that makes this possible can be used in conjunction with an arbitrary distance measure between input and output vectors. Thus, Hamming distance, Lee distance, or a burst distance can be assumed. Examples are presented for each of these distances.

Wainberg, S.↗
NASA NTRS1990

Performance comparison of combined ECC/RLL codes

In this paper, we present a performance comparison of several combined error correcting/run-lenth limited (ECC/RLL) codes created by concatenating a convolutional code with a run-length limited code. In each case, encoding and decoding are accomplished using a single trellis based on the combined code. Half of the codes under investigation use conventionally (d,k) run-length limited codes, where d is the minimum and k is the maximum allowable run of 0's between 1's. The other half of the combined codes use a special class of (d,k) codes known as distance preserving codes. These codes have the property that pairwise Hamming distances out of the (d,k) encoder are at least as large as the corresponding distances into the encoder (i.e., the codes preserve distance). Thus a combined code, created using a convolutional code concatenated with a distance preserving (d,k) code, will have a free distance (dfree) no smaller than the free distance of the original convolutional code. It should be noted that this does not hold if the (d,k) code was not distance preserving. A computer simulation is used to compare the performance of these two types of codes over the binary symmetric channel for various (d,k) constraints, rates, free distances, and numbers of states. Of particular interest for magnetic recording applications are codes with run-length constraints (1,3), (1,7), and (2,7).

French, C.↗
NASA NTRS1985

Codes with Parity Conditions on Subsets of Coordinates

Binary codes with the constraint that the codes restricted to certain subsets of columns must be contained in particular codes of the shorter lengths are considered. In particular, codes of even length 2k, and of minimum distance approximately greater than d, where in the code obtained by restricting to the first k positions has even weight and at the same time the code obtained by restricting to the last k positions also has even weight are considered. If k = 2n, n odd, and d = 2n, it is proved that the code has at most 8n - 4 codewords, and 8n - 4 is attainable for n = 3. This permits a file-transfer protocol control function assignment for personal computers to be chosen for 20 control functions using essentially just pairs of upper-case alphabetic ASCII characters where the Hamming distance between the binary forms of every two different control functions is at least six.

Posner, E. C.↗
NASA NTRS1989

Bandwidth efficient coding for fading channels - Code construction and performance analysis

The authors apply a general method of bounding the event error probability of trellis-coded modulation schemes to fading channels and use the effective length and the minimum-squared-product distance to replace the minimum-free-squared-Euclidean distance as code design parameters for Rayleigh and Rician fading channels with a substantial multipath component. They present 8-PSK trellis codes specifically constructed for fading channels that outperform equivalent codes designed for the additive white Gaussian noise channel when v is greater than or equal to 5. For quasiregular trellis codes there exists an efficient algorithm for evaluating event error probability, and numerical results on Pe which demonstrate the importance of the effective length as a code design parameter for fading channels with or without side information have been obtained. This is consistent with the case for binary signaling, where the Hamming distance remains the best code design parameter for fading channels. The authors show that the use of Reed-Solomon block codes with expanded signal sets becomes interesting only for large values of E(s)/N(0), where they begin to outperform trellis codes.

Schlegel, Christian↗
NASA NTRS1972

Error-erasure decoding of product codes.

Two error-erasure decoding algorithms for product codes that correct all the error-erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given. The first algorithm works when at least one of the component codes is majority-logic decodable. The second algorithm works for any product code. Both algorithms use the decoders of the component codes.

Wainberg, S.↗
NASA NTRS1974

Performance analysis of a frame sync algorithm for uncoded PSK telemetry

The optimum procedure for locating a frame sync word periodically inserted in uncoded binary data received over a binary symmetric channel is based on the Hamming distance metric. In this paper, a practical frame sync acquisition and maintenance algorithm is described, and its performance is analyzed. Specifically, with respect to this algorithm, an upper bound on the probability of false sync acquisition, the mean time to sync acquisition, and the subsequent mean time to loss of sync are computed for arbitrary bit error rates, frame lengths, sync word lengths, and algorithm parameters.

Levitt, B. K.↗
NASA NTRS1987

The capacity of the Hopfield associative memory

Techniques from coding theory are applied to study rigorously the capacity of the Hopfield associative memory. Such a memory stores n-tuple of + or - 1s. The components change depending on a hard-limited version of linear functions of all other components. With symmetric connections between components, a stable state is ultimately reached. By building up the connection matrix as a sum-of-outer products of m fundamental memories, it may be possible to recover a certain one of the m memories by using an initial n-tuple probe vector less than a Hamming distance n/2 away from the fundamental memory. If m fundamental memories are chosen at random, the maximum asymptotic value of m in order that most of the m original memories are exactly recoverable is n/(2 log n). With the added restriction that every one of the m fundamental memories be recoverable exactly, m can be no more than n/(4 log n) asymptotically as n approaches infinity. Extensions are also considered, in particular to capacity under quantization of the outer-product connection matrix. This quantized memory-capacity problem is closely related to the capacity of the quantized Gaussian channel.

Mceliece, Robert J.↗
NASA NTRS1988

Sparse distributed memory prototype: Principles of operation

Sparse distributed memory is a generalized random access memory (RAM) for long binary words. Such words can be written into and read from the memory, and they can be used to address the memory. The main attribute of the memory is sensitivity to similarity, meaning that a word can be read back not only by giving the original right address but also by giving one close to it as measured by the Hamming distance between addresses. Large memories of this kind are expected to have wide use in speech and scene analysis, in signal detection and verification, and in adaptive control of automated equipment. The memory can be realized as a simple, massively parallel computer. Digital technology has reached a point where building large memories is becoming practical. The research is aimed at resolving major design issues that have to be faced in building the memories. The design of a prototype memory with 256-bit addresses and from 8K to 128K locations for 256-bit words is described. A key aspect of the design is extensive use of dynamic RAM and other standard components.

Flynn, Michael J.↗
NASA NTRS1990

Two-dimensional shape recognition using sparse distributed memory

Researchers propose a method for recognizing two-dimensional shapes (hand-drawn characters, for example) with an associative memory. The method consists of two stages: first, the image is preprocessed to extract tangents to the contour of the shape; second, the set of tangents is converted to a long bit string for recognition with sparse distributed memory (SDM). SDM provides a simple, massively parallel architecture for an associative memory. Long bit vectors (256 to 1000 bits, for example) serve as both data and addresses to the memory, and patterns are grouped or classified according to similarity in Hamming distance. At the moment, tangents are extracted in a simple manner by progressively blurring the image and then using a Canny-type edge detector (Canny, 1986) to find edges at each stage of blurring. This results in a grid of tangents. While the technique used for obtaining the tangents is at present rather ad hoc, researchers plan to adopt an existing framework for extracting edge orientation information over a variety of resolutions, such as suggested by Watson (1987, 1983), Marr and Hildreth (1980), or Canny (1986).

Kanerva, Pentti↗
NASA NTRS1990

Depth-first search approach for fault-tolerant routing in hypercube multicomputers

Using depth-first search, the authors develop and analyze the performance of a routing scheme for hypercube multicomputers in the presence of an arbitrary number of faulty components. They derive an exact expression for the probability of routing messages by way of optimal paths (of length equal to the Hamming distance between the corresponding pair of nodes) from the source node to an obstructed node. The obstructed node is defined as the first node encountered by the message that finds no optimal path to the destination node. It is noted that the probability of routing messages over an optimal path between any two nodes is a special case of the present results and can be obtained by replacing the obstructed node with the destination node. Numerical examples are given to illustrate the results, and they show that, in the presence of component failures, depth-first search routing can route a message to its destination by means of an optimal path with a very high probability.

Chen, Ming-Syan↗
NASA NTRS1991

Associative-Memory Array Of Optical Logic Gates

Experimental optical associative-memory apparatus measures similarity or dissimilarity between input binary image and each of M binary images in memory. Uses Hamming distance as measure of dissimilarity, determines which (if any) of memory images closely or most closely resembles input image. Indicates this match by displaying image (e.g., recognized image or symbol, word, number, or other substitute image) in real time on output plane.

Chao, Tien-Hsin↗