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Uniform binary codes with special correlation including transorthogonality and simplex code, Hadamard matrices and difference sets uniform binary codes with special correlation including transorthogonality and simplex code, Hadamard matrices and difference sets
It was shown earlier that for a punctured Reed-Muller (RM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagram is much greater than that for an equivalent code obtained by a proper permutation on the bit positions. To find a permutation on the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is an interesting and challenging problem. This permutation problem is related to the generalized Hamming weight hierarchy of a code, and is shown that for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to Wei. The state complexity of trellis diagram for the extended and permuted (64, 24) BCH code is discussed.
A tabulation of binary systematic convolutional codes with an optimum distance profile for rates 1/3 and 1/4 is given. A number of short rate 1/3 binary nonsystematic convolutional codes are listed. These latter codes are simultaneously optimal for the following distance measures: distance profile, minimum distance, and free distance; they appear attractive for use with Viterbi decoders. Comparisons with previously known codes are made.
Error correcting binary block code decoding on Q-ary output channels /weighted erasure decoding/, considering applications
Using a simple combinatorial construction, the existence of a binary simplex code with m codewords for all m is greater than or equal to 1 is proved. The problem of the shortest possible length is left open.
Coding, using 3-bit binary words, is applicable to any measurement having integer scale up to 100. System using 6-bit data words can be expanded to read from 1 to 10,000, and 9-bit data words can increase range to 1,000,000. Code may be ''read'' directly by observation after memorizing simple listing of 9's and 10's.
We consider the problem of computing a sparse binary representation of an image. Given an image and an overcomplete, non-orthonormal basis, we aim to find a sparse binary vector indicating the minimal set of basis vectors that when added together best reconstruct the given input. We formulate this problem with an L 2 loss on the reconstruction error, and an L 0 loss on the binary vector enforcing sparsity. First, we solve the sparse representation QUBOs by solving them both on a D-Wave quantum annealer with Pegasus chip connectivity, as well as on the Intel Loihi 2 spiking neuromorphic processor using a stochastic Non-equilibrium Boltzmann Machine (NEBM). Second, using Quantum Evolution Monte Carlo with Reverse Annealing and iterated warm starting on Loihi 2 to evolve the solution quality from the respective machines. We demonstrate that both quantum annealing and neuromorphic computing are suitable for solving binary sparse coding QUBOs.
A computer program has been developed that analyzes Deep Space Network monitor data, looking for changes of trends in critical parameters. This program represents a significant improvement over the previous practice of manually plotting data and visually inspecting the resulting graphs to identify trends. This program uses proven numerical techniques to identify trends. When a statistically significant trend is detected, then it is characterized by means of a symbol that can be used by pre-existing model-based reasoning software. The program can perform any of the following functions: Given an expectation that data in a given list should exhibit an upward, downward, constant, or unknown trend, it can determine whether the data do or do not follow such a trend. Given a list of data, it can identify which of the aforementioned trends the data follow. Given two lists of data, it can determine whether or not both follow the same trend. This program can be executed on a variety of computers. It can be distributed in either source code or binary code form. It must be run in conjunction with any one of a number of Lisp compilers that are available commercially or as shareware.
A computer program defines and implements an innovative kind of data structure than can be used for representing information derived from serendipitous discoveries made via collection of scientific data on long exploratory spacecraft missions. Data structures capable of collecting any kind of data can easily be implemented in advance, but the task of designing a fixed and efficient data structure suitable for processing raw data into useful information and taking advantage of serendipitous scientific discovery is becoming increasingly difficult as missions go deeper into space. The present software eases the task by enabling definition of arbitrarily complex data structures that can adapt at run time as raw data are transformed into other types of information. This software runs on a variety of computers, and can be distributed in either source code or binary code form. It must be run in conjunction with any one of a number of Lisp compilers that are available commercially or as shareware. It has no specific memory requirements and depends upon the other software with which it is used. This program is implemented as a library that is called by, and becomes folded into, the other software with which it is used.
An application library package was developed that represents data packets for Deep Space Network (DSN) message packets as dynamically alterable arrays composed of arbitrary polymorphic data types. The software was to address a limitation of the present state of the practice for having an array directly composed of a single monomorphic data type. This is a severe limitation when one is dealing with science data in that the types of objects one is dealing with are typically not known in advance and, therefore, are dynamic in nature. The unique feature of this approach is that it enables one to define at run-time the dynamic shape of the matrix with the ability to store polymorphic data types in each of its indices. Existing languages such as C and C++ have the restriction that the shape of the array must be known in advance and each of its elements be a monomorphic data type that is strictly defined at compile-time. This program can be executed on a variety of platforms. It can be distributed in either source code or binary code form. It must be run in conjunction with any one of a number of Lisp compilers that are available commercially or as shareware.
We have performed numerical calculations of a binary interacting with a gas disk, using 11 different numerical methods and a standard binary-disk setup. The goal of this study is to determine whether all codes agree on a numerically converged solution and to determine the necessary resolution for convergence and the number of binary orbits that must be computed to reach an agreed-upon relaxed state of the binary-disk system. We find that all codes can agree on a converged solution (depending on the diagnostic being measured). The zone spacing required for most codes to reach a converged measurement of the torques applied to the binary by the disk is roughly 1% of the binary separation in the vicinity of the binary components. For our disk model to reach a relaxed state, codes must be run for at least 200 binary orbits, corresponding to about a viscous time for our parameters, 0.2(a 2 Ω B /v) is the kinematic viscosity. The largest discrepancies between codes resulted from the dimensionality of the setup (3D vs. 2D disks). We find good agreement in the total torque on the binary between codes, although the partition of this torque between the gravitational torque, orbital accretion torque, and spin accretion torque depends sensitively on the sink prescriptions employed. In agreement with previous studies, we find a modest difference in torques and accretion variability between 2D and 3D disk models. We find cavity precession rates to be appreciably faster in 3D than in 2D.
We present a novel data compression technique, called recursive interleaved entropy coding, that is based on recursive interleaving of variable-to-variable length binary source codes.
Virtually all previously-suggested rate 1/2 binary convolutional codes with KE = 24 are compared. Their distance properties are given; and their performance, both in computation and in error probability, with sequential decoding on the deep-space channel is determined by simulation. Recommendations are made both for the choice of a specific KE = 24 code as well as for codes to be included in future coding standards for the deep-space channel. A new result given in this report is a method for determining the statistical significance of error probability data when the error probability is so small that it is not feasible to perform enough decoding simulations to obtain more than a very small number of decoding errors.
A tabulation is given of long systematic and long quick-look-in (QLI) nonsystematic rate R = 1/2 binary convolutional codes with an optimum distance profile (ODP). These codes appear attractive for use with sequential decoders. Simulations for two of the new codes are reported and confirm Massey's conjecture that systematic and non-systematic codes of the same rate yield nearly identical computational and error probability performance with sequential decoding when the number of digits transmitted in the tail of the encoded frame is the same for both codes.
Three optimality criteria for convolutional codes are considered in this correspondence: namely, free distance, minimum distance, and distance profile. Here we report the results of computer searches for rate one-half binary convolutional codes that are 'robustly optimal' in the sense of being optimal for one criterion and optimal or near-optimal for the other two criteria. Comparisons with previously known codes are made. The results of a computer simulation are reported to show the importance of the distance profile to computational performance with sequential decoding.
Since 1971, when the Viterbi Algorithm was introduced as the optimal method of decoding convolutional codes, improvements in circuit technology, especially VLSI, have steadily increased its speed and practicality. Trellis-Coded Modulation (TCM) combines convolutional coding with higher level modulation (non-binary source alphabet) to provide forward error correction and spectral efficiency. For binary codes, the current stare-of-the-art is a 64-state Viterbi decoder on a single CMOS chip, operating at a data rate of 25 Mbps. Recently, there has been an interest in increasing the speed of the Viterbi Algorithm by improving the decoder architecture, or by reducing the algorithm itself. Designs employing new architectural techniques are now in existence, however these techniques are currently applied to simpler binary codes, not to TCM. The purpose of this report is to discuss TCM architectural considerations in general, and to present the design, at the logic gate level, or a specific TCM decoder which applies these considerations to achieve high-speed decoding.
The burst-b distance between two binary vectors is defined and shown to be a metric. This definition is applied to a binary-input, Q-ary output channel where errors occur in bursts. A decoding algorithm is presented for such a channel that is an extension of Weldon's (1971) weighted erasure decoding. Examples are presented illustrating the techniques.
Combinatorial formula for calculating coding vector sums - weight distribution moments of binary code - communication system