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IEEE Transactions on Information Theory

Soft-decision decoding using punctured codes

Ilya I. Dumer

Published Jan 1, 2001• 16 citations
Publisher

Abstract

Let a q-ary linear (n,k)-code be used over a memoryless channel. We design a soft-decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s /spl isin/ [k,n]. First, we take s cyclically consecutive positions starting from any initial point. Then we cut the subinterval of length s into two parts and examine T most plausible error patterns on either part. To obtain codewords of a punctured (s,k)-code, we try to match the syndromes of both parts. Finally, the designed codewords of an (s,k)-code are re-encoded to find the most probable codeword on the full length n. For any long linear code, the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum-likelihood (ML) decoding given sufficiently large T. By optimizing s, we prove that this near-ML decoding can be achieved by using only T/spl ap/q/sup (n-k)k/(n+k)/ error patterns. For most long linear codes, this optimization also gives about T re-encoded codewords. As a result, we obtain the lowest complexity order of q/sup (n-k)k/(n+k)/ known to date for near-ML decoding. For codes of rate 1/2, the new bound grows as a cubic root of the general trellis complexity q/sup min{n-k,k}/. For short blocks of length 63, the algorithm reduces the complexity of the trellis design by a few decimal orders.

DOI
10.1109/18.904512

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IEEE Transactions on Information Theory•Jan 1, 2000•31 citations
Let a q-ary linear (n,k)-code be used over a memoryless channel. We design a soft-decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s /spl isin/ [k,n]. First, we take s cyclically consecutive positions starting from any initial point. Then we cut the subinterval of length s into two parts and examine T most plausible error patterns on either part. To obtain codewords of a punctured (s,k)-code, we try to match the syndromes of both parts. Finally, the designed codewords of an (s,k)-code are re-encoded to find the most probable codeword on the full length n. For any long linear code, the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum-likelihood (ML) decoding given sufficiently large T. By optimizing s, we prove that this near-ML decoding can be achieved by using only T/spl ap/q/sup (n-k)k/(n+k)/ error patterns. For most long linear codes, this optimization also gives about T re-encoded codewords. As a result, we obtain the lowest complexity order of q/sup (n-k)k/(n+k)/ known to date for near-ML decoding. For codes of rate 1/2, the new bound grows as a cubic root of the general trellis complexity q/sup min{n-k,k}/. For short blocks of length 63, the algorithm reduces the complexity of the trellis design by a few decimal orders.
Let a q-ary linear (n, k) code C be used over a memoryless channel. We design a decoding algorithm /spl Psi//sub N/ that splits the received block into two halves in n different ways. First, about /spl radic/N error patterns are found on either half. Then the left- and right-hand lists are sorted out and matched to form codewords. Finally, the most probable codeword is chosen among at most n/spl radic/N codewords obtained in all n trials. The algorithm can be applied to any linear code C and has complexity order of n/sup 3//spl radic/N. For any N/spl ges/q/sup n-k/, the decoding error probability P/sub N/ exceeds at most 1+q/sup n-k//N times the probability P/sub /spl Psi//(C) of maximum-likelihood decoding. For code rates R/spl ges/1/2, the complexity order q/sup n-k/2/ grows as square root of general trellis complexity q/sup min{n-k,k}/. When used on quantized additive white Gaussian noise (AWGN) channels, the algorithm /spl Psi//sub N/ can provide maximum-likelihood decoding for most binary linear codes even when N has an exponential order of q/sup n-k/.
Suboptimal decoding of linear codes in "symmetric" memoryless channels is considered. For the q-ary codes of length n /spl rarr/ /spl infin/ and code rate R the number of decoding operations is upper bounded by the value q/sup n(c+o(1)), where o(1) /spl rarr/ 0 and c = min(R(1 - R), (1 - R)/2). The decoding error probability /spl epsi/ is upper bounded by the double error probability /spl epsi//sub e/ of maximum likelihood (ML) decoding, while /spl epsi/ /spl sim/ /spl epsi//sub e/, when n /spl rarr/ /spl infin/. For the channels with discrete (quantified) output the better estimate c = R(1 - R)/(1 + R) is obtained.
Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM (1, m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences wj|yj- cj| taken over all n positions. We obtain a tight upper bound LTon the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O (n ln2LT). In particular, all possible error weights wjequal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM (1, m) and requires O (n ln2n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4-1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed (0, ½), our algorithm outputs the complete list of codewords within the decoding radius n(½-) with linear complexity of order n ln2.
This paper presents a new low-weight trellis-based soft-decision iterative decoding algorithm for binary linear block codes. The algorithm is devised based on a set of optimality conditions and the generation of a sequence of candidate codewords for an optimality test. The initial candidate codeword is generated by a simple decoding method. The subsequent candidate codewords, if needed, are generated by a chain of low-weight trellis searches, one at a time. Each search is conducted through a low-weight trellis diagram centered around the latest candidate codeword and results in an improvement over the previous candidate codewords that have been already tested. This improvement is then used as the next candidate codeword for a test of optimality. The decoding iteration stops whenever a candidate codeword is found to satisfy a sufficient condition on optimality or the latest low-weight trellis search results in a repetition of a previously generated candidate codeword. A divide-and-conquer technique is also presented for codes that are not spanned by their minimum-weight codewords. The proposed decoding algorithm has been applied to some well-known codes of lengths 48, 64, and 128. Simulation results show that the proposed algorithm achieves either practically optimal error performance for the example codes of length 48 and 64 or near optimal error performance for the (128, 29, 32) RM code with a significant reduction in computational decoding complexity.
Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart
We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.