This letter presents a technique of combining multilevel coded modulation and product coding to form product modulation codes which achieve low bit error rates with reduced decoding complexity, Three multistage decodi...
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This letter presents a technique of combining multilevel coded modulation and product coding to form product modulation codes which achieve low bit error rates with reduced decoding complexity, Three multistage decoding algorithms are presented, and good codes for both the additive white Gaussian noise (AWGN) and Rayleigh fading channels have been constructed.
We describe an efficient algorithm for successive errors-and-erasures decoding of BCH codes, The decoding algorithm consists of finding all necessary error locator polynomials and errata evaluator polynomials, choosin...
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We describe an efficient algorithm for successive errors-and-erasures decoding of BCH codes, The decoding algorithm consists of finding all necessary error locator polynomials and errata evaluator polynomials, choosing the most appropriate error locator polynomial and errata evaluator polynomial, using these two polynomials to compute a candidate codeword for the decoder output, and testing the candidate for optimality,ia an originally developed acceptance criterion, Even in the most stringent case possible, the acceptance criterion is only a little more stringent than Forney's criterion for GMD decoding, We present simulation results on the error performance of our decoding algorithm for binary antipodal signals over an AWGN channel and a Rayleigh fading channel, The number of calculations of elements in a finite field that are required by our algorithm is only slightly greater than that required by hard-decision decoding, while error performance is almost as good as that achieved with GMD decoding, The presented algorithm is also applicable to efficient decoding of product RS codes.
A new bounded-distance (BD) decoding algorithm is presented for binary linear (n, k, d) block codes on additive white Gaussian noise channels, The algorithm is based on the generalized minimum distance (GMD) decoding ...
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A new bounded-distance (BD) decoding algorithm is presented for binary linear (n, k, d) block codes on additive white Gaussian noise channels, The algorithm is based on the generalized minimum distance (GMD) decoding algorithm of Forney using the acceptance criterion of Taipale and Pursley (GMD/TP), It is shown that the GMD/TP decoding algorithm is a BD decoding algorithm with effective error coefficient [GRAPHICS] It is also shown that the decision regions of GMD/TP are good inner approximations of those of full GMD decoding, and therefore full GMD decoding is BD and has an effective error coefficient that is well approximated by [GRAPHICS] Moreover, by adding a d-erasure-correction step to GMD decoding, the effective error coefficient can be reduced to A(d), the number of minimum-weight codewords, which is the same as the effective error coefficient of maximum-likelihood decoding, The decoding algorithm is mainly based on algebraic errors-and-erasures decoding and therefore has polynomial rather than exponential complexity.
It is shown that multistage generalized minimum-distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and complexity, We introduce a reliability metric for...
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It is shown that multistage generalized minimum-distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and complexity, We introduce a reliability metric for Gaussian channels that is easily computed from an inner product, and prove that a multistage GMD decoder using this metric is a bounded-distance decoder up to the true packing radius, The effective error coefficient of multistage GMD decoding is determined, Two simple modifications in the GMD decoding algorithm that drastically reduce this error coefficient are proposed, It is shown that with these modifications GMD decoding achieves the error coefficient of maximum-likelihood decoding for block codes and for generalized Construction A lattices. Multistage GMD decoding of the lattices D-4, E(8), K-12, BW16, and Lambda(24) is investigated in detail, For K-12, BW16, and Lambda(24), the GMD decoders have considerably lower complexity than the best known maximum-likelihood or bounded-distance decoding algorithms, and appear to be the most practically attractive decoders available, For high-dimensional codes and lattices (greater than or equal to 64 dimensions) maximum-likelihood decoding becomes infeasible, while GMD decoding algorithms remain quite practical, As an example, we devise a multistage GMD decoder for a 128-dimensional sphere packing with a nominal coding gain of 8.98 dB that attains an effective error coefficient of 1 365 760. This decoder requires only about 400 real operations, in addition to algebraic errors-and-erasures decoding of certain BCH and Hamming codes, It therefore appears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.
It has long been known that convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm [35], [11]. It has gradually become apparent that linear block ...
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It has long been known that convolutional codes have a natural, regular, trellis structure that facilitates the implementation of Viterbi's algorithm [35], [11]. It has gradually become apparent that linear block codes also have a natural, though not in general a regular, ''minimal'' trellis structure, which allows them to be decoded with a Viterbi-like algorithm [2], [36], [25], [12], [30], [16], [13], [18], [27], [28], [9], [17]. In both cases, the complexity of an unenhanced Viterbi decoding algorithm can be accurately estimated by the number of trellis edge symbols per encoded bit. It would therefore appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes [5], are known to have trellis representations which are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the ''minimal'' trellis representation. Thus ironically, we seem to know more about the minimal trellis representation for block than for convolutional codes. In this paper we provide a remedy, by developing a theory of minimal trellises for convolutional codes. (A similar theory has recently been given by Sidorenko and Zyablov [32].) This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-canonical generator matrix, from which the minimal trellis for the code can de directly constructed. Another by-product is that in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small.
In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algor...
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In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algortithm is Theta(e), This result suggests that the ''best'' trellis representation for a given linear block code is the one with the fewest edges, We will then show that, among all trellises that represent a given code, the original trellis introduced by BahI, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf, Massey, and Forney, uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine, we will also discuss ''trellis-oriented'' or ''minimal-span'' generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it.
As the codes that can prove Shannon's channel coding theorem by constructing the code, only two classes of codes are known, i.e., the concatenated code proposed by Forney and the iterated code proposed by Elias. I...
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As the codes that can prove Shannon's channel coding theorem by constructing the code, only two classes of codes are known, i.e., the concatenated code proposed by Forney and the iterated code proposed by Elias. In spite of this fact, those two codes have not been compared by the common evaluation measure. The former is evaluated from the standpoint of information theory and coding theory in terms of the average power from the viewpoint of the reliability function and the asymptotic distance ratio of the constructed code. For the latter, it is shown under a restricted condition that the decoding error rate per binary symbol converges to zero, only for the constructed code. With this as background, this paper compares the concatenated code and the iterated code by the common evaluation measure, considering the computational complexity required for the decoding. It is shown that the concatenated code is better than the iterated code. It is shown also that those two kinds of codes can be realized with the decoding complexity of polynomial order of the code length. Under the foregoing condition, it is shown that it is impossible to construct an iterated code with the nonzero asymptotic distance ratio. For the Justesen code, which is a class of the constructive concatenated codes, and the Elias iterated code, which is a class of the iterated codes, the upper bound for the decoding error probability is separated into a function of code length and a function of code rate. The code length is represented as a function of the computational complexity required for the decoding, and the upper bounds of the decoding error probabilities of the two codes are compared for the same computational complexity and the code rate. It is shown as a result that the upper bound for the decoding error probability of the Justesen code is asymptotically better than that of the Elias iterated code.
This paper deals with the decoding of the [23,12,7] binary Golay code. Recently, S.W. Wei and C.H. Wei suggested a step-by-step decoding algorithm. We present an improvement in the comparison circuit of their algorith...
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This paper deals with the decoding of the [23,12,7] binary Golay code. Recently, S.W. Wei and C.H. Wei suggested a step-by-step decoding algorithm. We present an improvement in the comparison circuit of their algorithm and construct a very high-speed parallel decoder.
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