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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.
This paper suggests guidelines for the utilization of low-redundancy convolutional codes with emphasis on providing a quick look capability (no decoding) and a moderate amount of coding gain. The performance and implementation complexity of threshold, Viterbi, and sequential decoding when used with low-redundancy, systematic, convolutional codes is discussed. An extensive list of optimum, short constraint length codes is found for use with Viterbi decoding, and several good, long constraint length codes are found for use with sequential decoding.
Results of research on the use of convolutional codes in data communications are presented. Convolutional coding fundamentals are discussed along with modulation and coding interaction. Concatenated coding systems and data compression with convolutional codes are described.
The performance of convolutional codes in fading channels typical of the planetary entry channel is examined in detail. The signal fading is due primarily to turbulent atmospheric scattering of the RF signal transmitted from an entry probe through a planetary atmosphere. Short constraint length convolutional codes are considered in conjunction with binary phase-shift keyed modulation and Viterbi maximum likelihood decoding, and for longer constraint length codes sequential decoding utilizing both the Fano and Zigangirov-Jelinek (ZJ) algorithms are considered. Careful consideration is given to the modeling of the channel in terms of a few meaningful parameters which can be correlated closely with theoretical propagation studies. For short constraint length codes the bit error probability performance was investigated as a function of E sub b/N sub o parameterized by the fading channel parameters. For longer constraint length codes the effect was examined of the fading channel parameters on the computational requirements of both the Fano and ZJ algorithms. The effects of simple block interleaving in combatting the memory of the channel is explored, using the analytic approach or digital computer simulation.
A method of using error-correcting codes to obtain data compression, called syndrome-source-coding, is described in which the source sequence is treated as an error pattern whose syndrome forms the compressed data. It is shown that syndrome-source-coding can achieve arbitrarily small distortion with the number of compressed digits per source digit arbitrarily close to the entropy of a binary memoryless source. A universal generalization of syndrome-source-coding is formulated which provides robustly-effective, distortionless, coding of source ensembles.
The performance of convolutional codes in fading channels typical of the planetary entry channel is examined in detail. Short constraint length codes are considered in conjunction with binary phase-shift-keyed (BPSK) modulation and Viterbi maximum likelihood decoding while for longer constraint length codes we consider sequential decoding utilizing both the Fano and Zigangirov-Jelinek (ZJ) algorithms. For short constraint length codes we are primarily interested in the bit error probability performance parameterized by the fading channel parameters. For longer constraint length codes interest will center on the effect of the fading channel parameters on the computational requirements of both the Fano and ZJ algorithms. In either case the effects of simple block interleaving in combatting the memory of the channel is thoroughly explored. The approach is analytic where possible otherwise resort is made to digital computer simulation.
A class of codes defined by the parity check matrix of a linear code of minimum distance at least t + 1 is examined and the number of parity check symbols is discussed. Determinants of this type are known as alternants (Muir and Metzler, 1930); codes corresponding to such a matrix are termed alternant codes. The parity check matrix is obtained by restricting some of its elements to subfields of GF(q to the m-th power). Minimum distance and redundancy bounds are established for these codes, and some interesting equivalence and invariance properties are derived.
Automated code evaluation system can be used to detect coding errors and unsound coding practices in any ANSI FORTRAN IV source code before they can cause execution-time malfunctions. System concentrates on acceptable FORTRAN code features which are likely to produce undesirable results.
For coded telemetry systems it is important to know the tradeoff between the error probability and the complexity of implementation. For systems using block codes, the block coding error exponent is a good way to estimate this tradeoff. The new upper bounds on the minimum distance of binary codes result in improved upper bounds on the coding error exponents for binary input memoryless channels.
A digital computer code CAVE (Conduction Analysis Via Eigenvalues), which finds application in the analysis of two dimensional transient heating of hypersonic vehicles is described. The CAVE is written in FORTRAN 4 and is operational on both IBM 360-67 and CDC 6600 computers. The method of solution is a hybrid analytical numerical technique that is inherently stable permitting large time steps even with the best of conductors having the finest of mesh size. The aerodynamic heating boundary conditions are calculated by the code based on the input flight trajectory or can optionally be calculated external to the code and then entered as input data. The code computes the network conduction and convection links, as well as capacitance values, given basic geometrical and mesh sizes, for four generations (leading edges, cooled panels, X-24C structure and slabs). Input and output formats are presented and explained. Sample problems are included. A brief summary of the hybrid analytical-numerical technique, which utilizes eigenvalues (thermal frequencies) and eigenvectors (thermal mode vectors) is given along with aerodynamic heating equations that have been incorporated in the code and flow charts.
The general problem of image data compression is discussed briefly with attention given to the use of Karhunen-Loeve transforms, suboptimal systems, and block quantization. A survey is then conducted encompassing the four categories of adaptive systems: (1) adaptive transform coding (adaptive sampling, adaptive quantization, etc.), (2) adaptive predictive coding (adaptive delta modulation, adaptive DPCM encoding, etc.), (3) adaptive cluster coding (blob algorithms and the multispectral cluster coding technique), and (4) adaptive entropy coding.
As part of the continuing effort to introduce standardization of spacecraft and ground equipment in satellite systems, NASA's Goddard Space Flight Center and other NASA facilities have supported the development of a set of standards for the use of error control coding in telemetry subsystems. These standards are intended to ensure compatibility between spacecraft and ground encoding equipment, while allowing sufficient flexibility to meet all anticipated mission requirements. The standards which have been developed to date cover the application of block codes in error detection and error correction modes, as well as short and long constraint length convolutional codes decoded via the Viterbi and sequential decoding algorithms, respectively. Included are detailed specifications of the codes, and their implementation. Current effort is directed toward the development of standards covering channels with burst noise characteristics, channels with feedback, and code concatenation.
CELFE (Coupled Eulerian Lagrangian Finite Element)/NASTRAN Code three-dimensional finite element code has the capability for analyzing of structures subjected to high velocity impact. The local response is predicted by CELFE and, for large problems, the far-field impact response is predicted by NASTRAN. The coupling of the CELFE code with NASTRAN (CELFE/NASTRAN code) and the application of the code to selected three-dimensional high velocity impact problems are described.
The CELFE (Coupled Eulerian Lagrangian Finite Element)/NASTRAN Code three-dimensional finite element code has the capability for analyzing of structures subjected to high velocity impact. The local response is predicted by CELFE and, for large problems, the far-field impact response is predicted by NASTRAN. The coupling of the CELFE code with NASTRAN (CELFE/NASTRAN code) and the application of the code to selected three-dimensional high velocity impact problems are described.
Using a general decoding technique of Solomon we evaluate the performance of certain block codes on a Gaussian channel. Quadratic residue codes of lengths 48 and 80 as well as BCH codes of length 128 and rates 1/2 and 1/3 are considered. All four of these codes perform quite favorably with respect to the constraint-length 7 rate 1/2 convolutional code presently used on NASA's Mariner-class spacecraft.
A computer oriented time code designed for users with various time resolution requirements is presented. It is intended as a time code for spacecraft and ground applications where direct code compatibility with automatic data processing equipment is of primary consideration. The principal features of this time code are: byte oriented format, selectable resolution options (from seconds to nanoseconds); and long ambiguity period. The time code is compatible with the new data handling and management concepts such as the NASA End-to-End Data System and the Telemetry Data Packetization format.
The synchronization capabilities of Reed-Solomon codes when an appropriate coset of the code is used instead of the code itself are examined. In this case an E-error correcting Reed-Solomon code is transformed into a code capable of determining that there are m symbols out of sync, if e symbol errors occurred, whenever m + e E. In the event that m = 0, i.e., the word is in sync, then decoder will correct any pattern of E - 1 on fewer symbol errors.
Computer simulation results are presented on the performance of convolutional codes of constraint lengths 7 and 10 concatenated with the (255, 223) Reed-Solomon code (a proposed NASA standard). These results indicate that as much as 0.8 dB can be gained by concatenating this Reed-Solomon code with a (10, 1/3) convolutional code, instead of the (7, 1/2) code currently used by the DSN.