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{{Expert
In [[coding theory]], '''decoding''' is the process of translating received messages into [[codewords]] of a given [[code]]. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a [[noisy channel]], such as a [[binary symmetric channel]].
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Upon fixing <math>\mathbb{P}(x \mbox{ received})</math>, <math>x</math> is restructured and
<math>\mathbb{P}(y \mbox{ sent})</math> is constant as all codewords are equally likely to be sent.
Therefore,
<math>
\mathbb{P}(x \mbox{ received} \mid y \mbox{ sent})
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Note that this is already of significantly less complexity than that of a [[standard array|standard array decoding]].
However, under the assumption that no more than <math>t</math> errors were made during transmission, the receiver can look up the value <math>He</math> in a further reduced table of size
:<math>
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</math>
only (for a binary code). The table is against pre-computed values of <math>He</math> for all possible error patterns <math>e \in \mathbb{F}_2^n</math>.
Knowing what <math>e</math> is, it is then trivial to decode <math>x</math> as:
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* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | year=1992 | isbn=3-540-54894-7 }}
==
{{reflist}}
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