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Replication decoding

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3 Author(s)

Any symbol in a redundant code can be recovered when it belongs to certain erasure patterns. Several alternative expressions of a given symbol, to be referred to as its replicas, can therefore be computed in terms of other ones. Decoding is interpreted as decoding upon a received symbol, given itself and a number of such replicas, expressed in terms of other received symbols. For linear q-ary (n,k) block codes, soft-decision demodulation and memoryless channels, the maximum-likelihood decision rule on a given symbol is formulated in terms of r \leq n - k linearly independent replicas from the parity-check equations. All replicas deriving from the r selected replicas by linear combination are actually taken into account in this decision rule. Its implementation can be direct; use transformations or a sequential circuit implementing a trellis representation of the parity-check matrix. If r = n - k , decoding is optimum, in the sense of symbol-by-symbol maximum-likelihoed. Simplification results in the transformed and sequential implementations when r < n - k . If the selected replicas are disjoint, generalized ( q -ary, weighted) threshold decoding results. The decoding process can easily be modffied in order to provide word-by-word maximum-likelihood decoding. Convolutional codes are briefly considered. Two specific problems are discussed: the use of previous decisions, which leads to a weighted generalization of feedback decoding, and the extension of replication decoding to nonsystematic codes.

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IEEE Transactions on Information Theory  (Volume:25 ,  Issue: 3 )