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IEEE Information Theory Workshop (ITW)

Box and match soft decision decoding of linear block codes with iterative information set reduction

M.P.C. Fossorier, A. Valembois

Published Aug 27, 2003• 6 citations
Publisher

Abstract

The order statistic decoding (OSD) algorithm is a probabilistic list decoding algorithm which allows to achieve practically optimum soft decision decoding of binary linear block codes of length up to 128. Recently, matching techniques were applied to the OSD algorithm to reduce both the worst case and average complexities of decoding at the expense of memory. The corresponding box-and-match algorithm (BMA) allows to achieve near optimum decoding of codes of length up to 192. In this work, we investigate the application of iterative information set reduction to the BMA, which provides further refinements in the trade-offs between error performance and decoding complexity.

DOI
10.1109/itw.2002.1115412

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•Jul 10, 1996•7 citations
The order statistic decoding (OSD) algorithm is a probabilistic list decoding algorithm which allows to achieve practically optimum soft decision decoding of binary linear block codes of length up to 128. Recently, matching techniques were applied to the OSD algorithm to reduce both the worst case and average complexities of decoding at the expense of memory. The corresponding box-and-match algorithm (BMA) allows to achieve near optimum decoding of codes of length up to 192. In this work, we investigate the application of iterative information set reduction to the BMA, which provides further refinements in the trade-offs between error performance and decoding complexity.
In this paper, we improve the ordered statistics decoding algorithm by using matching techniques. This allows us: to reduce the worst case complexity of decoding (the error performance being fixed) or to improve the error performance (for a same complexity); to reduce the ratio between average complexity and worst case complexity; to achieve practically optimal decoding of rate-1/2 codes of lengths up to 128 (rate-1/2 codes are a traditional benchmark, for coding rates different from 1/2, the decoding is easier); to achieve near-optimal decoding of a rate-1/2 code of length 192, which could never be performed before.
In this paper, efficient methods to improve the box and matching algorithm (BMA) are presented. Firstly, an efficient approach is introduced to terminate the decoding if a local optimal candidate satisfies a probabilistic sufficient condition. The false alarm probability associated with the use of the probabilistic sufficient condition is also derived. Secondly, by constructing a control band which is assumed error free, the matching capability of the BMA is enhanced. More precisely, the performance of BMA of order (i + 1) is nearly achieved with a small increase in complexity and no increase in memory with respect to the BMA of order i. A tight performance analysis is derived based on the theory of order statistics. An error floor associated either with false alarms or with errors in the control band is introduced, but this error floor can be controlled using the analysis in both cases. Simulation results show that the performance of the enhanced BMA for the decoding of the RS(255,239) code with BPSK signaling over an AWGN channel is about 0.1 dB away from that of maximum likelihood decoding at the word error rate (WER) 10-3.
The ordered statistics-based list decoding techniques for linear binary block codes of small to medium block length are investigated. The construction of a list of the test error patterns is considered. The original ordered-statistics decoding (OSD) is generalized by assuming segmentation of the most reliable independent positions (MRIPs) of the received bits. The segmentation is shown to overcome several drawbacks of the original OSD. The complexity of the ordered statistics-based decoding is further reduced by assuming a partial ordering of the received bits in order to avoid the complex Gauss elimination. The probability of the test error patterns in the decoding list is derived. The bit error rate performance and the decoding complexity trade-off of the proposed decoding algorithms is studied by computer simulations. Numerical examples show that, in some cases, the proposed decoding schemes are superior to the original OSD in terms of both the bit error rate performance as well as the decoding complexity.
This dissertation presents a novel approach to soft decision decoding for binary linear block codes. The basic idea of this approach is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is highly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the trade-off between performance and decoding complexity is provided. The proposed decoding is based on the reordering of the received symbols according to their reliability measure. In the dissertation, we evaluate the statistics of the noise after ordering. Based on these statistics, we derive two monotonic properties which dictate the reprocessing strategy. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test highly related to the reprocessing strategy is introduced to reduce the number of computations for each reprocessing stage. For short codes of lengths N ≤ 32 or medium codes with 32 < N ≤ 64 with rate R ≥ 0.6, near-optimum bit error performance is achieved in two stages of reprocessing with at most a computation complexity of o(K2) constructed code-words, where K is the dimension of the code. For longer codes, three or more reprocessing stages are required to achieve near-optimum decoding. However, most of the coding gain is obtained within the first two reprocessing stages for error performances of practical interest. The proposed decoding algorithm applies to any binary linear code, does not require any data storage and is well suitable for parallel processing. Furthermore, the maximum number of computations required at each reprocessing stage is fixed, which prevents buffer overflow at low SNR. Further reductions of the number of computations based on probabilistic or decomposable properties of the code considered are also investigated. The generalization of the algorithm to other channels is discussed. Finally, the discrete time channel associated with the AWGN model after ordering is greatly simplified. This simplification allows an important reduction of the number of computations required to evaluate the theoretical error performance of any algorithm based on a partial or total ordering.
Soft-decision decoding of a linear block code using the most reliable basis corresponding to each received word is investigated. Based either on probabilistic properties or on the structure of the code considered, three improvements to the algorithm devised by Fossorier and Lin (see ibid., vol.41, no.9, p.1379-1396, 1995) are presented. These modifications allow large computation savings or significant decoding speedup with little error performance degradation. First, a reduced probabilistic list of codeword candidates is associated with order-i reprocessing of a given code. It results in a large reduction of the maximum number of computations with a very small degradation in performance. Then, a probabilistic stopping criterion is introduced for order-0 reprocessing. This new test significantly decreases the average number of computations when appropriately implemented. Finally, the application of the algorithm to coset decoding is considered for |u|u+v| constructed codes. In addition to the conventional coset decoding, a new adaptive practically optimum coset decoding method is presented where at each reprocessing stage, the number of surviving cosets decreases. Suboptimum closest coset decoding is also investigated. It is shown that two-stage decoding with the algorithm of Fossorier and Lin offers a large variety of choices, since the reprocessing order of each stage can be determined independently.