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IEEE International Symposium on Information Theory (ISIT)

On Survivor Error Patterns For Maximum Likelihood Soft Decoding

Jakov Snyders

Published Aug 24, 2005• 6 citations

Abstract

Summary Consider an (t1.k.d) binary linear block code specified by a parity-check matrix H. Assume that codewords with equal proba- bility are transmitted through a niemoryless channel. Let Y be the bit-by-bit hard-detected version of the received word, and assume that z, the corresponding syndrome, is nonzero. A set of 1 linearly independent columns of H that adds up to z will be called a /-pat- tern. Thus t does not exceed ni where n1 = ti-k is the number of check bits. A palier'ri is a I-pattern with unspecified 1. Assume, without essential loss of generality, that d t 3. Let r be the set of columns of H; then IF( = ri due to the aforementioned assumption. The ivei,oh/ of a column h of H is defined to be the confidence value (magnitude of the log likelihood ratio) of the bit associated with h. The weigh/ of a subset @ of r is the sum of the weights of the elements of @. An algorithm (I) for carrying out maximum likelihood decoding may now be phrased as follows: I) among all the /-patterns, with 1 ranging up to ni, find (the usually unique) one with least weight, then

DOI: 10.1109/isit.1991.695248

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Summary Consider an (t1.k.d) binary linear block code specified by a parity-check matrix H. Assume that codewords with equal proba- bility are transmitted through a niemoryless channel. Let Y be the bit-by-bit hard-detected version of the received word, and assume that z, the corresponding syndrome, is nonzero. A set of 1 linearly independent columns of H that adds up to z will be called a /-pat- tern. Thus t does not exceed ni where n1 = ti-k is the number of check bits. A palier'ri is a I-pattern with unspecified 1. Assume, without essential loss of generality, that d t 3. Let r be the set of columns of H; then IF( = ri due to the aforementioned assumption. The ivei,oh/ of a column h of H is defined to be the confidence value (magnitude of the log likelihood ratio) of the bit associated with h. The weigh/ of a subset @ of r is the sum of the weights of the elements of @. An algorithm (I) for carrying out maximum likelihood decoding may now be phrased as follows: I) among all the /-patterns, with 1 ranging up to ni, find (the usually unique) one with least weight, then

With any parity check matrix H of a code C(n,k,d) one may associate a graph whose vertices represent distinct bases, i.e., sets of n-k independent columns of H. Such basis specifies a unique error pattern. A search is carried out over a limited part of the bases graph for an error pattern having the minimum confidence value. Properties of the bases graph are utilized to reduce the complexity of the search procedure. Bounds are derived for the worst case computational complexity as well as for the correct-decoding probability.

A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed numberj \geq 3of l's and each row contains a small fixed numberk > jof l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixedj. When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixedj. A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. Forj > 3and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.

In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.

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.