We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo-soft decision decoding can be extended from one threshold T to z>1 thresholds T_11 thresholds case, starting with a geometric interpretation of the optimal threshold location problem and using an approach from Zyablov.
We consider the Additive White Gaussian Noise channel with Binary Phase Shift Keying modulation. Our aim is to enable an algebraic hard decision Bounded Minimum Distance decoder for a binary block code to exploit soft information obtained from the demodulator. This idea goes back to Forney and is based on treating received symbols with low reliability as erasures. This erasing at the decoder is done using a threshold, each received symbol with reliability falling below the threshold is erased. Depending on the target overall complexity of the decoder this pseudo-soft decision decoding can be extended from one threshold T to z>1 thresholds T_11 thresholds case, starting with a geometric interpretation of the optimal threshold location problem and using an approach from Zyablov.
In this correspondence, we consider the decoding of binary block codes over the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) signaling. By a reliability-order-based decoding algorithm (ROBDA), we mean a soft-decision decoding algorithm which decodes to the best (most likely) codeword of the form that is the sum of the hard-decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions. Examples of ROBDAs include many well-known decoding algorithms, such as the generalized-minimum-distance (GMD) decoding algorithm, Chase decoding algorithms, and the reliability-based decoding algorithms proposed by Fossorier and Lin. It is known that the squared error-correction-radii of ROBDAs can be computed from the minimal squared Euclidean distances (MSEDs) between the all-one sequence and the polyhedra corresponding to the error patterns. For the computation of such MSEDs, we give a new method which is more compact than the one proposed by Fossorier and Lin. These results are further used to show that any bounded-distance ROBDA is asymptotically optimal: The ratio between the probability of decoding error of a bounded-distance ROBDA and that of the maximum-likelihood (ML) decoding approaches 1 when the signal-to-noise ratio (SNR) approaches infinity, provided that the minimum Hamming distance of the code is greater than 2.
The error-correction capability and the performance of generalized minimum distance decod- ing based on threshold erasing is studied as a function of the number of decoding trials and of the allowable reliability values. Generalized Minimum Distance (GMD) decoding, as intro- duced by Forney (l), is a multitrial decoding algorithm that combines reliability information, represented by numbers in the closed interval (0,1), with algebraic decoding algorithms for error correction. In each trial, a fixed number of the most unreliable symbols in the received sequence is erased. If a cer- tain distance criterion is satisfied, Forney has shown that the transmitted codeword can be retrieved in at most (d/2) trials, where d is the Hamming distance of the code. In this work, we study a variation on Forney's scheme where, in each trial, symbols whose reliabilities are below a certain threshold are erased, and hence there is no need to sort the received symbols according to reliability values be- fore erasing as in Forney's GMD decoding. Furthermore, we consider limiting the allowable reliability values to the set R, = u~=mo/2'{1 - 2i/m}, where m = 1,2,3,. . .. This covers the simple cases of taking only hard decisions (m = l), allow- ing erasures (m = 2), or accommodating the full range (0,1) of reliability values as in Forney's GMD decoding (m + 00). Limiting the reliability values may be of importance since, in any digital implementation, the reliability values need to be quantized. Moreover, in some applications, e.g., GMD decod- ing of concatenated codes over discrete channels, the reliabil- ity values are discrete in nature (2). The maximum number of trials is kept as a parameter. Limiting the reliability values and the number of trials leads to the concept of reduced GMD decoding.
Bit-by-bit soft-decision decoding of binary cyclic codes is considered. A significant reduction in decoder complexity can be achieved by requiring only that the decoder correct all analog error patterns which fall within a Euclidean sphere whose radius is equal to half the minimum Euclidean distance of the code. Such a "maximum-radius" scheme is asymptotically optimum for the additive white Gaussian noise (AWGN) channel. An iterative extension of the basic algebraic analog decoding scheme is discussed, and performance curves are given for the (17,9), (21,11), and (73,45) codes on the AWGN channel.
Soft decision decoding of binary linear block codes transmitted over the additive white Gaussian channel (AWGN) using antipodal signaling is considered. A set of decoding algorithms called generalized Chase algorithms is proposed. In contrast to Chase algorithms, which require a\lfloor (d- 1)/2 \rfloorbinary error-correcting decoder for decoding a binary linear block code of minimum distanced, the generalized Chase algorithms can use a binary decoder that can correct less than\lfloor ( d - 1)/2 \rfloorhard errors. The Chase algorithms are particular cases of the generalized Chase algorithms. The performance of all proposed algorithms is asymptotically optimum for high signal-to-noise ratio (SNR). Simulation results for the(47, 23)quadratic residue code indicate that even for low SNR the performance level of a maximum likelihood decoder can be approached by a relatively simple decoding procedure.
This dissertation develops a set of new algorithms for soft decision decoding of binary linear block codes. The performance (i.e. the probability of error) of these algorithms is asymptotically optimum for high signal-to-noise ratio for binary linear block codes transmitted over the additive white gaussian (AWGN) channel using antipodal signalling.
Among the algorithms proposed are the generalized Chase algorithms. In contrast to Chase algorithms, which require a {(d-1)/2} hard error-correcting decoder for decoding a binary linear block code of minimum distance d, the generalized Chase algorithms can use a binary decoder that can correct less than {(d-1)/2} hard errors. The Chase algorithms are particular cases of the generalized Chase algorithms. Modified generalized Chase algorithm 2 and 3 are proposed for codes that have an overall parity check. These algorithms use fewer test patterns than the generalized Chase algorithms and they also have asymptotically optimum performance for high signal-to-noise ratio.
The erasure information decoding algorithm of Tanaka and Kakigahara is generalized so that any hard decoder that corrects only up to t errors, 0 (LESSTHEQ) t (LESSTHEQ) {(d-1)/2} can be used to obtain asymptotically optimal performance for high signal-to-noise ratio. The modified erasure information decoding algorithm is proposed for codes that have an overall parity check. The modified erasure information decoding algorithm requires fewer test patterns than the erasure information decoding algorithm and its performance is also asymptotically optimum for high signal-to-noise ratio.
The algorithms proposed by Dorsch and, Baumert and McEliece are analyzed to determine the conditions under which their performance would be asymptotically optimum for high signal-to-noise ratio.
Simulation results for the (47,23) Quadratic Residue code are shown to demonstrate the performance of the algorithms. The results indicate that even for low signal-to-noise ratio the performance of a maximum likelihood decoder can be approached by a relatively simple decoding procedure.
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 (1989) using the acceptance criterion of Taipale and Pursley (GMD/TP) proposed in 1991. It is shown that the GMD/TP decoding algorithm is a BD decoding algorithm with effective error coefficient (/sup n//sub d/). 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 (/sup n//sub d/). Moreover, by adding a d-erasure-correction step to GMD decoding, the effective error coefficient can be reduced to A/sub 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.