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The Consultative committee for Space Data STandards (CCSDS) recommends that space communication links employ a concatenated error-correcting channel-coding system in which the inner code is a convolutional (7, 2/2) code and the outer code is a (255,223) Reed-Solomon code.
A robust error-control coding system is presented. This system is a cascaded FEC (forward error control) scheme supported by parity retransmissions for further error correction in the erroneous data words. The error performance and throughput efficiency of the system are analyzed. Two specific examples of the error-control system are studied. The first example does not use an inner code, and the outer code, which is not interleaved, is a shortened code of the NASA standard RS code over GF(28). The second example, as proposed for NASA, uses the same shortened RS code as the base outer code C2, except that it is interleaved to a depth of 2. It is shown that both examples provide high reliability and throughput efficiency even for high channel bit-error rates in the range of 0.01.
The performance of maximum-distance-separable codes with symbols from GF(1) when they are used for pure error detection or for simultaneous error correction and detection is investigated. These codes are good for symbol error detection. Their probability of undetected error is upper bounded by q to the -r and decreases monotonically as the symbol error probability epsilon decreases from (q-1)/q to 0, where r is the number of parity-check symbols of a code.
The performance of maximum-distance-separable codes with symbols from GF(1) when they are used for pure error detection or for simultaneous error correction and detection is investigated. These codes are good for symbol error detection. Their probability of undetected error is upper bounded by q to the -r and decreases monotonically as the symbol error probability epsilon decreases from (q-1)/q to 0, where r is the number of parity-check symbols of a code.
N, k group codes without repeated columns obtained by puncturing coordinates of length shift register code
A decoding method for a (23,12) Golay code is extended to the important 1/2-rate (24,12) Golay code so that three errors can be corrected and four errors can be detected. It is shown that the method can be extended to any decoding method which can correct three errors in the (23,12) Golay code.
A self-synchronizing coding scheme for NASA's TDRSS satellite system is a concatenation of a (2,1,7) inner convolutional code with a (255,223) Reed-Solomon outer code. Both symbol and word synchronization are achieved without requiring that any additional symbols be transmitted. An important parameter which determines the performance of the word sync procedure is the ratio of the decoding failure probability to the undetected error probability. Ideally, the former should be as small as possible compared to the latter when the error correcting capability of the code is exceeded. A computer simulation of a (255,223) Reed-Solomon code as carried out. Results for decoding failure probability and for undetected error probability are tabulated and compared.
A computer code is now available for estimating the error in results from the STAGS finite element code for a shell unit consisting of a rectangular orthotropic plate. This memorandum contains basic information about the computer code EAC (Error Analysis and Correction) and describes the connection between the input data for the STAGS shell units and the input data necessary to run the error analysis code. The STAGS code returns a set of nodal displacements and a discrete set of stress resultants; the EAC code returns a continuous solution for displacements and stress resultants. The continuous solution is defined by a set of generalized coordinates computed in EAC. The theory and the assumptions that determine the continuous solution are also outlined in this memorandum. An example of application of the code is presented and instructions on its usage on the Cyber and the VAX machines have been provided.
A method for high speed decoding a d sub min = 6 Reed-Solomon (RS) code is presented. Properties of the two byte error correcting and three byte error detecting RS code are discussed. Decoding using a quadratic equation is shown. Theorems and concomitant proofs are included to substantiate this decoding method.
A method has been developed of computing bounds on the performance of a code comprised of two linear binary codes generated by two encoders serially concatenated through an interleaver. Originally intended for use in evaluating the performances of some codes proposed for deep-space communication links, the method can also be used in evaluating the performances of short-block-length codes in other applications. The method applies, more specifically, to a communication system in which following processes take place: At the transmitter, the original binary information that one seeks to transmit is first processed by an encoder into an outer code (Co) characterized by, among other things, a pair of numbers (n,k), where n (n > k)is the total number of code bits associated with k information bits and n k bits are used for correcting or at least detecting errors. Next, the outer code is processed through either a block or a convolutional interleaver. In the block interleaver, the words of the outer code are processed in blocks of I words. In the convolutional interleaver, the interleaving operation is performed bit-wise in N rows with delays that are multiples of B bits. The output of the interleaver is processed through a second encoder to obtain an inner code (Ci) characterized by (ni,ki). The output of the inner code is transmitted over an additive-white-Gaussian- noise channel characterized by a symbol signal-to-noise ratio (SNR) Es/No and a bit SNR Eb/No. At the receiver, an inner decoder generates estimates of bits. Depending on whether a block or a convolutional interleaver is used at the transmitter, the sequence of estimated bits is processed through a block or a convolutional de-interleaver, respectively, to obtain estimates of code words. Then the estimates of the code words are processed through an outer decoder, which generates estimates of the original information along with flags indicating which estimates are presumed to be correct and which are found to be erroneous. From the perspective of the present method, the topic of major interest is the performance of the communication system as quantified in the word-error rate and the undetected-error rate as functions of the SNRs and the total latency of the interleaver and inner code. The method is embodied in equations that describe bounds on these functions. Throughout the derivation of the equations that embody the method, it is assumed that the decoder for the outer code corrects any error pattern of t or fewer errors, detects any error pattern of s or fewer errors, may detect some error patterns of more than s errors, and does not correct any patterns of more than t errors. Because a mathematically complete description of the equations that embody the method and of the derivation of the equations would greatly exceed the space available for this article, it must suffice to summarize by reporting that the derivation includes consideration of several complex issues, including relationships between latency and memory requirements for block and convolutional codes, burst error statistics, enumeration of error-event intersections, and effects of different interleaving depths. In a demonstration, the method was used to calculate bounds on the performances of several communication systems, each based on serial concatenation of a (63,56) expurgated Hamming code with a convolutional inner code through a convolutional interleaver. The bounds calculated by use of the method were compared with results of numerical simulations of performances of the systems to show the regions where the bounds are tight (see figure).
Self correcting codes for structural redundancy and reliability of automata
The distance structure of cyclic codes of composite length was investigated. A lower bound on the minimum distance for this class of codes is derived. In many cases, the lower bound gives the true minimum distance of a code. Then the distance structure of the direct sum of two cyclic codes of composite length were investigated. It was shown that, under certain conditions, the direct-sum code provides two levels of error correcting capability, and hence is a two-level unequal error protection (UEP) code. Finally, a class of two-level UEP cyclic direct-sum codes and a decoding algorithm for a subclass of these codes are presented.
Two highly reliable codes termed "Modified b-adjacent interleaving codes" provide fail-safe operation of launch processing and control system in which common memory is coordination point for interconnection of up to 64 minicomputers. Codes detect and correct bit errors in computer data transmission.
NASA Lewis Research Center s Communications Technology Division has an ongoing program in the development of efficient channel coding schemes for satellite communications applications. Through a university grant, as a part of this research, the University of Toledo is investigating the performance of turbocodes, which use parallel concatenation of non-systematic convolutional encoders with an interleaver. The error correcting capacity of these codes is close to the Shannon limit. The research emphasis is on the development of low-complexity, but higher rate (greater than one half), turbocodes and on the iterative decoding of block codes.
In a complex computer environment there is ample opportunity for error, a mistake by a programmer, or a software-induced undesirable side effect. In insurance, errors can cost a company heavily, so protection against inadvertent change is a must for the efficient firm. The data processing center at Transport Life Insurance Company has taken a step to guard against accidental changes by adopting a software package called EQNINT (Equations Interpreter Program). EQNINT cross checks the basic formulas in a program against the formulas that make up the major production system. EQNINT assures that formulas are coded correctly and helps catch errors before they affect the customer service or its profitability.
The paper deals with a method developed for decoding a (255, 223) Reed-Solomon code over GF(2 to the 8th) with both errors and erasures. The matrix of decoding times for correcting errors and erasures of the code using a simplified decoder is presented. It is shown that the algorithm proposed is faster by a factor of from three to seven.
Coding interpretation, examining rook packing, latin squares, affine planes and error distributing codes
STAR computer has five redundant modular function units, fixed store, arithmetic, memory, input, and output. Each unit is connected to a diagnostic control unit, each is coded for error detection and error correction. Separation into function units permits assembly of many different systems from the set of units.