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NASA NTRS1976

The fast decoding of Reed-Solomon codes using number theoretic transforms

It is shown that Reed-Solomon (RS) codes can be encoded and decoded by using a fast Fourier transform (FFT) algorithm over finite fields. The arithmetic utilized to perform these transforms requires only integer additions, circular shifts and a minimum number of integer multiplications. The computing time of this transform encoder-decoder for RS codes is less than the time of the standard method for RS codes. More generally, the field GF(q) is also considered, where q is a prime of the form K x 2 to the nth power + 1 and K and n are integers. GF(q) can be used to decode very long RS codes by an efficient FFT algorithm with an improvement in the number of symbols. It is shown that a radix-8 FFT algorithm over GF(q squared) can be utilized to encode and decode very long RS codes with a large number of symbols. For eight symbols in GF(q squared), this transform over GF(q squared) can be made simpler than any other known number theoretic transform with a similar capability. Of special interest is the decoding of a 16-tuple RS code with four errors.

Reed, I. S.↗

NASA NTRS1986

Optimum coding techniques for MST radars

The optimum coding technique for MST (mesosphere stratosphere troposphere) radars is that which gives the lowest possible sidelobes in practice and can be implemented without too much computing power. Coding techniques are described in Farley (1985). A technique mentioned briefly there but not fully developed and not in general use is discussed here. This is decoding by means of a filter which is not matched to the transmitted waveform, in order to reduce sidelobes below the level obtained with a matched filter. This is the first part of the technique discussed here; the second part consists of measuring the transmitted waveform and using it as the basis for the decoding filter, thus reducing errors due to imperfections in the transmitter. There are two limitations to this technique. The first is a small loss in signal to noise ratio (SNR), which usually is not significant. The second problem is related to incomplete information received at the lowest ranges. An appendix shows a technique for handling this problem. Finally, it is shown that the use of complementary codes on transmission and nonmatched decoding gives the lowest possible sidelobe level and the minimum loss in SNR due to mismatch.

Sulzer, M. P.↗

NASA NTRS1983

New Syndrome Decoding Techniques for the (n, K) Convolutional Codes

This paper presents a new syndrome decoding algorithm for the (n,k) convolutional codes (CC) which differs completely from an earlier syndrome decoding algorithm of Schalkwijk and Vinck. The new algorithm is based on the general solution of the syndrome equation, a linear Diophantine equation for the error polynomial vector E(D). The set of Diophantine solutions is a coset of the CC. In this error coset a recursive, Viterbi-like algorithm is developed to find the minimum weight error vector (circumflex)E(D). An example, illustrating the new decoding algorithm, is given for the binary nonsystemmatic (3,1)CC.

Reed, I. S.↗

NASA NTRS1984

New syndrome decoding techniques for the (n, k) convolutional codes

This paper presents a new syndrome decoding algorithm for the (n, k) convolutional codes (CC) which differs completely from an earlier syndrome decoding algorithm of Schalkwijk and Vinck. The new algorithm is based on the general solution of the syndrome equation, a linear Diophantine equation for the error polynomial vector E(D). The set of Diophantine solutions is a coset of the CC. In this error coset a recursive, Viterbi-like algorithm is developed to find the minimum weight error vector (circumflex)E(D). An example, illustrating the new decoding algorithm, is given for the binary nonsystemmatic (3, 1)CC. Previously announced in STAR as N83-34964

Reed, I. S.↗

NASA NTRS1977

Minimax decoding of cyclic block codes

A minimax decoding algorithm utilizing soft bit detection of an (n,k) cyclic block code is described which will permit the correction of up to n-k bit errors interspersed at random locations throughout the block. The decoding solution consists of: (1) identifying the ordered soft bit set and, (2) finding the minimum order solution to the resulting syndrome equations where the nonzero error vector components are constrained to be a subset of the soft bit set. An efficient implementation of the decoding operation is described. In essence, this algorithm focuses the correction capability of the code on those bit positions which have the lowest a posteriori probabilities of correct detection.

Greene, E. P.↗

NASA NTRS1992

Rate 8/9 coded 8-PSK system for downlink applications

An advanced Coded Trellis Modulation (CTM) System which achieves a 2 bits/s/Hz bandwidth efficiency at an information rate of 200 Mbit/s while minimizing satellite power requirements, was developed for downlink earth station applications. The CTM system employs a high-speed rate 8/9 convolutional code with Viterbi decoding and an 8-Phase Shift Keying (PSK) modem. The minimum Euclidean distance between the modulated waveforms corresponding to the information sequences are maximized in order to maximize the noise immunity of the system. Nyquist filters with a square-root of 40 percent roll-off are used at the transmit and receive sides of the modem in order to minimize intersymbol interference, adjacent channel interference, and distortion at the nonlinear satellite power amplifier. The use of a coded system here also minimizes the effects of co-channel interference. The developed performance of the hardware system was measured to achieve within 1.5 dB from theory at a bit error rate of 5 x 10(exp -7) over an additive white Gaussian noise channel.

Fang, Russell↗

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 NTRS1983

Decoding: Codes and hardware implementation

The MST radars vary considerably from one installation to the next in the type of hardware, operating schedule and associated personnel. Most such systems do not have the computing power to decode in software when the decoding must be performed for each received pulse, as is required for certain sets of phase codes. These sets provide the best signal to sidelobe ratio when operating at the minimum band length allowed by the bandwidth of the transmitter. The development of the hardware phase decoder, and the applicability of each to decoding MST radar signals are discussed. A new design for a decoder which is very inexpensive to build, easy to add to an existing system and is capable of decoding on each received pulse using codes with a band length as short as one microsecond is presented.

Sulzer, M. P.↗

NASA NTRS1990

On decoding of multi-level MPSK modulation codes

The decoding problem of multi-level block modulation codes is investigated. The hardware design of soft-decision Viterbi decoder for some short length 8-PSK block modulation codes is presented. An effective way to reduce the hardware complexity of the decoder by reducing the branch metric and path metric, using a non-uniform floating-point to integer mapping scheme, is proposed and discussed. The simulation results of the design are presented. The multi-stage decoding (MSD) of multi-level modulation codes is also investigated. The cases of soft-decision and hard-decision MSD are considered and their performance are evaluated for several codes of different lengths and different minimum squared Euclidean distances. It is shown that the soft-decision MSD reduces the decoding complexity drastically and it is suboptimum. The hard-decision MSD further simplifies the decoding while still maintaining a reasonable coding gain over the uncoded system, if the component codes are chosen properly. Finally, some basic 3-level 8-PSK modulation codes using BCH codes as component codes are constructed and their coding gains are found for hard decision multistage decoding.

Lin, Shu↗

NASA NTRS1973

Data multiplexer using a tree switch

Self-decoding FET-hybrid or integrated-circuit tree configuration uses minimum number of components and can be sequenced by clock or computer. Redundancy features can readily be incorporated into tree configuration; as tree grows in size and more sensors are included, percentage of parts that will affect given percentage of sensors steadily decreases.

Easton, R. A.↗

NASA NTRS1977

Some rate 1/3 and 1/4 binary convolutional codes with an optimum distance profile

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.

Johannesson, R.↗

NASA NTRS1983

Simplified Syndrome Decoding of (n, 1) Convolutional Codes

A new syndrome decoding algorithm for the (n, 1) convolutional codes (CC) that is different and simpler than the previous syndrome decoding algorithm of Schalkwijk and Vinck is presented. The new algorithm uses the general solution of the polynomial linear Diophantine equation for the error polynomial vector E(D). This set of Diophantine solutions is a coset of the CC space. A recursive or Viterbi-like algorithm is developed to find the minimum weight error vector cirumflex E(D) in this error coset. An example illustrating the new decoding algorithm is given for the binary nonsymmetric (2,1)CC.

I. S. Reed↗

NASA NTRS1984

Sequential Syndrome Decoding of Convolutional Codes

The algebraic structure of convolutional codes are reviewed and sequential syndrome decoding is applied to those codes. These concepts are then used to realize by example actual sequential decoding, using the stack algorithm. The Fano metric for use in sequential decoding is modified so that it can be utilized to sequentially find the minimum weight error sequence.

Reed, I. S.↗

NASA NTRS1976

Differential pulse code modulation

A differential pulse code modulation (DPCM) encoding and decoding method is described along with an apparatus which is capable of transmission with minimum bandwidth. The apparatus is not affected by data transition density, requires no direct current (DC) response of the transmission link, and suffers from minimal ambiguity in resolution of the digital data.

Herman, C. F.↗

NASA NTRS1990

Multi-level modulation codes and multi-stage decoding

Multi-level method is a powerful technique for constructing bandwidth efficient modulation codes. It allows the construction of modulation codes systematically with arbitrary large minimum squared Euclidean distance from component codes in conjunction with proper bits-to-signal mapping. If the component codes are chosen properly, the resultant modulation code not only has good minimum squared Euclidean distance but is also rich in structural properties such as: linear structure, phase invariant property, and trellis structure. A modulation code with linear structure has invariant distance distribution. Phase invariant property is useful in resolving carrier-phase ambiguity and ensuring rapid carrier-phase resynchronization after temporary loss of synchronization. It the component codes have trellis structure, the resultant multi-level modulation code also has trellis structure. Trellis structure allows decoding of a multi-level modulation code with the soft-decision Viterbi decoding algorithm. Furthermore, the multi-level structure allows decoding of a multi-level modulation code with the multi-stage decoding. This type of decoding reduces the decoding complexity. Multi-stage decoding is not optimum even though the decoding of each component is optimum. Based on the analysis and simulation results, the difference in error performance between the optimum decoding of the overall multi-level modulation code and the suboptimum multi-stage decoding of the code is very little, a fraction of dB loss.

Lin, Shu↗

NASA NTRS1988

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme for error control in data communications is presented. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is particularly effective for high speed satellite communication for large file transfer where high reliability is required. Also presented is a simple method for constructing block codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft decision Viterbi decoding algorithm.

Kasami, Tadao↗

NASA NTRS1988

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme for error control in data communications is presented. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is very effective for high speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45 deg rotation.

Lin, Shu↗

NASA NTRS1988

A concatenated coded modulation scheme for error control (addition 2)

A concatenated coded modulation scheme for error control in data communications is described. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is particularly effective for high-speed satellite communications for large file transfer where high reliability is required. This paper also presents a simple method for constructing block codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft-decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45 deg rotation.

Lin, Shu↗