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IEEE Information Theory Workshop (ITW)

Information-set soft-decision decoding

Ilya I. Dumer

Published Nov 27, 2002• 3 citations
Publisher

Abstract

Information-set decoding is considered for soft decision channels. The sets of most plausible error patterns are represented as ellipsoids in Hamming spaces, and their minimum coverings are designed by random search. For general linear codes, maximum likelihood decoding with the lowest known complexity is obtained.
DOI
10.1109/itw.1998.706443

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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.
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/.