Replication decoding

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q-ary (n,k)</tex> block codes, soft-decision demodulation and memoryless channels, the maximum-likelihood decision rule on a given symbol is formulated in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r \leqn - k</tex> linearly independent replicas from the parity-check equations. All replicas deriving from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r = n - k</tex> , decoding is optimum, in the sense of symbol-by-symbol maximum-likelihoed. Simplification results in the transformed and sequential implementations when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r &lt; n - k</tex> . If the selected replicas are disjoint, generalized ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -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.