Showing papers on "Turbo code published in 1989"
27 Nov 1989
TL;DR: The Viterbi algorithm is modified to deliver the most likely path sequence in a finite-state Markov chain, as well as either the a posteriori probability for each bit or a reliability value, with the aim of producing soft decisions to be used in the decoding of outer codes.
Abstract: The Viterbi algorithm (VA) is modified to deliver the most likely path sequence in a finite-state Markov chain, as well as either the a posteriori probability for each bit or a reliability value. With this reliability indicator the modified VA produces soft decisions to be used in the decoding of outer codes. The inner software output Viterbi algorithm (SOVA) accepts and delivers soft sample values and can be regraded as a device for improving the signal-to-noise ratio, similar to an FM demodulator. Several applications are investigated to show the gain over the conventional hard-deciding VA, including concatenated convolutional codes, concatenation of trellis-coded modulation with convolutional FEC (forward error correcting) codes, and coded Viterbi equalization. For these applications additional gains of 1-4 dB as compared to the classical hard-deciding algorithms were found. For comparison, the more complex symbol-to-symbol MAP, whose optimal a posteriori probabilities can be transformed into soft outputs, was investigated. >
1,848 citations
TL;DR: An investigation is conducted of the high-rate punctured convolutional codes suitable for Viterbi and sequential decoding of known short-memory codes.
Abstract: An investigation is conducted of the high-rate punctured convolutional codes suitable for Viterbi and sequential decoding. Results on known short-memory codes M >
312 citations
TL;DR: This pragmatic approach to all coding applications permits the use of a single basic coder and decoder to achieve respectable coding (power) gains for bandwidth efficiencies from 1 b/s/Hz to 6 b/S/Hz.
Abstract: Since the early 1970s, for power-limited applications, the convolutional code constraint length K=7 and rate 1/2, optimum in the sense of maximum free distance and minimum number of bit errors caused by remerging paths at the free distance, has become the de facto standard for coded digital communication. This was reinforced when punctured versions of this code became the standard for rate 3/4 and 7/8 codes for moderately bandlimited channels. Methods are described for using the same K=7, rate 1/2 convolutional code with signal phase constellations of 8-PSK and 160PSK and quadrature amplitude constellations of 16-QASK, 64-QASK, and 256-QASK to achieve, respectively, 2 and 3, and 2, 4, and 6 b/s/Hz bandwidth efficiencies while providing power efficiency that in most cases is virtually equivalent to that of the best Ungerboeck codes for constraint length 7 or 64 states. This pragmatic approach to all coding applications permits the use of a single basic coder and decoder to achieve respectable coding (power) gains for bandwidth efficiencies from 1 b/s/Hz to 6 b/s/Hz. >
308 citations
TL;DR: An algorithm to compute the distance spectrum of linear, regular, and quasiregular trellis codes is presented, derived from a bidirectional stack algorithm, although it could also be based on the Viterbi algorithm.
Abstract: A class of quasiregular codesis defined for which the distance spectrum can be calculated from the codeword corresponding to the all-zero information sequence. Convolutional codes and regular codes are both quasiregular, as well as most of the best known trellis codes. An algorithm to compute the distance spectrum of linear, regular, and quasiregular trellis codes is presented. In particular, it can calculate the weight spectrum of convolutional (linear trellis) codes and the distance spectrum of most of the best known trellis codes. The codes do not have to be linear or regular, and the signals do not have to be used with equal probabilities. The algorithm is derived from a bidirectional stack algorithm, although it could also be based on the Viterbi algorithm. The algorithm is used to calculate the beginning of the distance spectrum of some of the best known trellis codes and to compute tight estimates on the first-event-error probability and on the bit-error probability. >
136 citations
TL;DR: A fast algorithm for searching a tree (FAST) is presented for computing the distance spectrum of convolutional codes and several of the listed encoders have distance spectra superior to those of any previously known codes of the same rate and memory.
Abstract: A fast algorithm for searching a tree (FAST) is presented for computing the distance spectrum of convolutional codes. The distance profile of a code is used to limit substantially the error patterns that have to be searched. The algorithm can easily be modified to determine the number of nonzero information bits of an incorrect path as well as the length of an error event. For testing systematic codes, a faster version of the algorithm is given. FAST is much faster than the standard bidirectional search. On a microVAX, d/sub infinity /=27 was verified for a rate R=1/2, memory M=25 code in 37 s of CPU time. Extensive tables of rate R=1/2 encoders are given. Several of the listed encoders have distance spectra superior to those of any previously known codes of the same rate and memory. A conjecture than an R=1/2 systematic convolutional code of memory 2M will perform as well as a nonsystematic convolutional code of memory M is given strong support. >
130 citations
Book•
31 May 1989
TL;DR: A review of Vector Spaces, the Division Algorithm and the Euclidean Algorithm, and field Representations and Zech's Log Tables.
Abstract: 1 Introduction and Fundamentals.- 2 Finite Fields.- 3 Linear Codes.- 4 Some Special Linear Codes.- Chapters 5 Cyclic Codes.- 6 BCH Codes and Bounds for Cyclic Codes.- 7 Error Correction Techniques and Digital Audio Recording.- A: Review of Vector Spaces.- B: The Division Algorithm and the Euclidean Algorithm.- C: The Chinese Remainder Theorem.- D: Field Representations and Zech's Log Tables.- References.
126 citations
TL;DR: An approach to private-key cryptosystems is proposed which allows use of very simple codes of distance, and it is shown that this approach can be used to solve the challenge of verifying identity of Turing-complete systems.
Abstract: An approach to private-key cryptosystems is proposed which allows use of very simple codes of distance >
78 citations
27 Nov 1989
TL;DR: Two generalized Viterbi algorithms (GVAs) for the decoding of convolutional codes are presented, a parallel algorithm that simultaneously identifies the L best estimates of the transmitted sequence, and a serial algorithm that identifies the lth best estimate using the knowledge about the previously found l-1 estimates.
Abstract: Presented are two generalized Viterbi algorithms (GVAs) for the decoding of convolutional codes. They are respectively, a parallel algorithm that simultaneously identifies the L best estimates of the transmitted sequence, and a serial algorithm that identifies the lth best estimate using the knowledge about the previously found l-1 estimates. These algorithms are applied to combined speech and channel coding systems, concatenated codes, trellis-coded modulation, partial response (continuous-phase modulation), and hybrid ARQ (automatic repeat request) schemes. As an example, for a concatenated code more than 2 dB is gained by the use of the GVA with L=3 over the Viterbi algorithm for block error rates less than 10/sup -2/. The channel is a Rayleigh fading channel. >
73 citations
02 Apr 1989
TL;DR: The author presents a number of spherical codes which can be implemented in practice and which are capable of providing significant coding gains on the AWGN channel and provides an alternative design method to the conventional algebraic approach.
Abstract: The author presents a number of spherical codes which can be implemented in practice and which are capable of providing significant coding gains on the AWGN channel. The geometrical approach to code design used also provides an alternative design method to the conventional algebraic approach. Other codes could easily be designed along similar lines, for a desired code rate or for some other desired characteristic. The principles given will help in the design of optimal codes. The approach has the additional advantage that it results in codes with a high degree of symmetry and a well-defined structure, which may be helpful in the design of a decoder.
60 citations
TL;DR: The genie-aided algorithm is used as a tool in estimating the asymptotic behavior of the M-algorithm and the results conform closely to experience with convolutional codes due to the similar distance structure of the codes.
Abstract: Comparisons are made of a genie-aided sequential algorithm due to D. Haccoun and M.J. Ferguson (1975), the Viterbi algorithm, the M-algorithm, and the Fano algorithm for rate-1/2 and rate-2/3 trellis modulation codes on rectangular signal sets. The effects of signal-to-noise ratio and decoding-delay constraints on the choice of decoding algorithms for framed data are examined by computer simulation. Additionally, the genie-aided algorithm is used as a tool in estimating the asymptotic behavior of the M-algorithm. In general, the results conform closely to experience with convolutional codes due to the similar distance structure of the codes. The Fano algorithm produces good error performance with a low average number of computations when long decoding delay is permissible. The M-algorithm provides a savings in computation compared to the Viterbi algorithm if a small decoding delay is required. >
TL;DR: An upper bound on the free distance of punctured codes is derived, indicating that puncture codes are good codes, and a construction method that generates the low-rate original codes which duplicate given known high-rate codes through perforation is proposed.
Abstract: The authors present some properties of punctured convolutional codes, providing a construction method and a list of new, good, high-rate, long-memory punctured codes. The structure of punctured codes is examined and an upper bound on the free distance of punctured codes is derived, indicating that punctured codes are good codes. A construction method that generates the low-rate original codes which duplicate given known high-rate codes through perforation is proposed. Tables of punctured codes that duplicate the best known nonsystematic codes of rates 2/3 and 3/4 with memory lengths ranging from 3 to 23 and from 3 to 9, respectively, are given, together with the best known systematic codes for rates ranging from 2/3 to 7/8 with very long memory, M=44 through 48. >
TL;DR: It is shown empirically that the number of good periodically time-varying convolutional codes increases exponentially with the period for any set of parameters, which means they can be used to enhance the security of cryptosystems without sacrificing error correction capability.
Abstract: It is shown empirically that the number of good periodically time-varying convolutional codes increases exponentially with the period for any set of parameters. Hence, they can be used to enhance the security of cryptosystems without sacrificing error correction capability. It is shown that some periodically time-varying convolutional codes improve the free distance compared with fixed codes. >
TL;DR: A class of convolutional codes called cross parity check (CPC) codes, which are useful for the protection of data stored on magnetic tape, is described and analyzed and the technique of parity check matrix reduction, which is useful for reducing the inherent decoding delay of CPC codes is described.
Abstract: A class of convolutional codes called cross parity check (CPC) codes, which are useful for the protection of data stored on magnetic tape, is described and analyzed. CPC codes are first explained geometrically; their construction is described in terms of constraining data written onto a tape in such a way that when lines of varying slope are drawn across the tape, the bits falling on those lines sum to zero modulo two. This geometric interpretation is then formalized by the construction of canonical parity check matrices and systematic generator matrices for CPC codes and by computing their constraint lengths. The distance properties of CPC codes are analyzed, and it is shown that these codes are maximum distance separable convolutional codes. In addition, examples are given of both error and erasure decoding algorithms that take advantage of the geometric regularity of CPC codes. The technique of parity check matrix reduction, which is useful for reducing the inherent decoding delay of CPC codes, is described. The technique consists of dividing each term of the parity check matrix by some polynomial and retaining only the remainder. A class of polynomials that are particularly attractive for this purpose if identified. >
28 Mar 1989
TL;DR: It is shown how distance preserving RLL codes can be used with binary convolutional codes to create combined ECC/RLL codes with the distance properties of the original Convolutional code.
Abstract: A subset of the RLL (run-length limited) codes called distance preserving RLL codes is introduced. In addition to satisfying a run-length constraint, these codes have the property that the Hamming distance between any two encoder output sequences is at least as large as the Hamming distance between the corresponding encoder input sequences. Thus, when used in combination with a binary ECC code, a distance preserving RL code does not reduce the overall Hamming distance of the ECC/RLL combination to something below the Hamming distance of the ECC code alone. It is shown how distance preserving RLL codes can be used with binary convolutional codes to create combined ECC/RLL codes with the distance properties of the original convolutional code. >
TL;DR: The initial vector problem is considered for group codes for the Gaussian channel and, in particular, for codes generated by permutation groups.
Abstract: The initial vector problem is considered for group codes for the Gaussian channel and, in particular, for codes generated by permutation groups. Variant I and variant II permutation codes are defined, and optimal initial vectors for these codes are constructed. A numerical algorithm for constructing optimal group codes is presented together with examples of this algorithm applied to several permutation groups. >
TL;DR: The results show that the symbol-oriented nature of multidimensional inner codes can provide an improvement of up to 1 dB in the overall performance of a concatenated coding system when these codes replace bit-oriented two-dimensional codes.
Abstract: A concatenated coding system using two-dimensional trellis-coded MPSK inner codes and Reed-Solomon outer codes for application in high-speed satellite communication systems was proposed previously by the authors (ibid., vol.37, no.5, p.420-7, May 1989). The authors extend their results to systems using symbol-oriented, multidimensional, trellis-coded MPSK inner codes. The concatenated coding systems are divided into two classes according to their achievable effective information rates. The first class uses multidimensional trellis-coded 8-PSK inner codes and achieves effective information rates around 1 b/dimension (spectral efficiency 2 b/s/Hz). The second class employs multidimensional trellis-coded 16-PSK inner codes and provides effective information rates around 1.5 b/dimension (spectral efficiency 3 b/s/Hz). Both classes provide significant coding gains over an uncoded reference system with the same effective information rate as the coded system. The results show that the symbol-oriented nature of multidimensional inner codes can provide an improvement of up to 1 dB in the overall performance of a concatenated coding system when these codes replace bit-oriented two-dimensional codes. >
TL;DR: Codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated, and necessary and sufficient conditions for the spectral properties are given.
Abstract: Codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated, and necessary and sufficient conditions for the spectral properties are given. In particular, for the most common balanced alphabets, a complete characterization is furnished for the class of symbol-by-symbol codes with vanishing spectrum and second derivative. The fundamental results are extended to block codes, and suitable criteria for their design are given. Specific examples of binary and ternary block codes are presented; their performance in terms of low power content at low frequencies compares advantageously with that of some conventional codes. >
01 Feb 1989
TL;DR: A coding scheme, based on product codes, is introduced, and the improvement of the code rate is demonstrated, compared to similar fixed rate codes.
Abstract: Owing to the time-varying nature of the HF channel, error correcting codes of a fixed rate provide unnecessary correction power for much of the time, with consequently low data rates. The purpose of an adaptive coding scheme is to permit only the necessary degree of error correction to be applied to the transmitted information according to the channel conditions. Such a coding scheme, based on product codes, is introduced, and the improvement, compared to similar fixed rate codes, is demonstrated. A method of improving the code rate further is described, and the results of this and an interleaving procedure are also given.
TL;DR: Tables summarizing some results on the size of optimal unidirectional error-correcting codes which follow from upper bounds on thesize of a code of length n correcting t or fewer uniddirectional errors are given.
Abstract: A brief introduction is given on the theory of codes correcting unidirectional errors, in the context of symmetric and asymmetric error-correcting codes. Upper bounds on the size of a code of length n correcting t or fewer unidirectional errors are then derived. Methods in which codes correcting up to t unidirectional errors are constructed by expurgating t-fold asymmetric error-correcting codes or by expurgating and puncturing t-fold symmetric error-correcting codes are also presented. Finally, tables summarizing some results on the size of optimal unidirectional error-correcting codes which follow from these bounds and constructions are given. >
TL;DR: The attributes of the several line codes considered for a North American standard for ISDN basic access are compared and an intuitive explanation of why the code performs as it does is attempted.
Abstract: The attributes of the several line codes considered for a North American standard for ISDN basic access are compared. Impairments that affect line code performance are examined, and their performance characteristics are discussed. An intuitive explanation of why the code performs as it does is attempted in each case. Some variations of the codes that were considered and that were intended to improve their performance are also discussed. The codes considered are bi-phase, modified duobinary, alternate mark inversion, 3B2T, 2B1Q and 4B3T. >
TL;DR: The author presents separable codes which can detect t or all unidirectional errors in VLSI circuits.
Abstract: Systematic and separable codes are widely used for error detection in VLSI circuits. Recently, systematic codes have been found for detecting t-unidirectional and burst unidirectional errors. In such codes, r checkbits are added to k information bits to form a codeword of length (k+r), and it is assumed that all 2/sup k/ possible information k-tuples are present. However, in many applications all 2/sup k/ possible k-tuples are not used. In such cases it is possible to obtain codes usually referred to as separable codes. The author presents separable codes which can detect t or all unidirectional errors. >
TL;DR: Algorithms for conversion from feedforward (FF) to feedback (FB) codes and vice versa are presented and are based on a trellis search and an algebraic solution.
Abstract: Algorithms for conversion from feedforward (FF) to feedback (FB) codes and vice versa are presented. The former is based on a trellis search and the latter on an algebraic solution. The algorithms work on trellis codes of rate R = k/n.
TL;DR: This paper gives a new interpretation to q-ary codes by utilizing the concept of superimposed concatenated codes and derives a lower bound on the minimum distance of q-ARY expanded RS codes over GF(qm) is derived and a new decoding algorithm is clarified.
Abstract: Error-correcting techniques are important in digital communications, for example, satellite communications, mobile communications, and digital audio systems. In digital communications, block codes such as Reed-Solomon (RS) codes (recommended for compact disc (CD) and digital audio tape (DAT)) are used.
This paper treats q-ary expanded RS codes over GF(qm) as two-dimensional codes over GF(q) and gives a new interpretation to q-ary codes by utilizing the concept of superimposed concatenated codes. Using this new interpretation a lower bound on the minimum distance of q-ary expanded RS codes over GF(qm) is derived and a new decoding algorithm is clarified.
01 Nov 1989
TL;DR: The author considers a concatenated coding scheme for digital recording systems that uses an inner code to detect errors during the storage/retrieval process and to erase the corresponding errors.
Abstract: The author considers a concatenated coding scheme for digital recording systems. This concatenated scheme uses an inner code to detect errors during the storage/retrieval process and to erase the corresponding errors. The Reed-Solomon code is used as an outer code to correct the errors (undetected errors) and the erasures (detected errors) generated by the inner code. Several error detecting codes are considered as the inner code, and the corresponding block error probabilities and average net information that can be stored and retrieved without error in a unit time are computed to make performance comparisons. >
02 Apr 1989
TL;DR: A set of error correcting line codes that are derived from known linear error control codes (ECCs) are presented and their performance aspects are analysed.
Abstract: A set of error correcting line codes (ECCLs) that are derived from known linear error control codes (ECCs) are presented and their performance aspects analysed. Special consideration has been given to the line coding characteristics, in particular spectral considerations, disparity and runlength bounds since the error control characteristics are inherited directly from the ECC. Results are given for various different codes and presented in tabular form.
TL;DR: A study of reduced complexity concatenated coding schemes, for commercial digital satellite systems with low-cost earth terminals, explored trade-offs between coding gain, overall rate and decoder complexity, and compared concatenation schemes with single codes.
Abstract: A study of reduced complexity concatenated coding schemes, for commercial digital satellite systems with low-cost earth terminals, is reported. The study explored trade-offs between coding gain, overall rate and decoder complexity, and compared concatenated schemes with single codes. It concentrated on short block and constraint length inner codes, with soft decision decoding, concatenated with a range of Reed-Solomon outer codes. The dimension of the inner code was matched to the outer code symbol size, and appropriate interleaving between the inner and outer codes was used. Very useful coding gains were achieved with relatively high-rate, low-complexity schemes. For example, concatenating the soft decision decoded (9,8) single parity check inner code with the CCSDS recommended standard Reed-Solomon outer code gives a coding gain of 4.8dB at a bit error probability of 10−5, with an overall rate of 0-78.
21 Jun 1989
TL;DR: A class of balanced (or DC-free) codes that are useful for unidirectional and asymmetric error detection is defined and optimal (serial and parallel) balanced codes are constructed.
Abstract: A class of balanced (or DC-free) codes that are useful for unidirectional and asymmetric error detection is defined. Optimal (serial and parallel) balanced codes are constructed. >
27 Nov 1989
TL;DR: The authors investigate the underlying statistical structure of the block error detection, correction, and misdetection events of a block code on a discrete renewal inner channel, such as a fading RF channel, and show that these events can be represented with outer channel models, i.e. partitioned Markov processes, similar to the inner channel models.
Abstract: The authors investigate the underlying statistical structure of the block error detection, correction, and misdetection events of a block code on a discrete renewal inner channel, such as a fading RF channel. They show that these events can be represented with outer channel models, i.e. partitioned Markov processes, similar to the inner channel models. Analytical procedures to determine the parameters of these outer channel models are presented. The application of these models in the design of such error-control systems as ARQ (automatic repeat request) systems is investigated. Consideration is also given to the memory-efficient recording of the channel error stream to be recorded or stored, in order to parameterize inner channel models. The results of several simulations to verify the analytical procedures are also presented. >
IBM1
TL;DR: The free distance of a convolutional code of rate 1/n is bounded by n times the constant length of its encoder to form a bound for all sufficiently large n.
Abstract: The free distance of a convolutional code of rate 1/n is bounded by n times the constant length of its encoder. Two classes of convolutional codes whose free distances meet this bound are studied. A technique of constructing convolutional codes that meet this bound for all sufficiently large n is given. >