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This article discusses common methods in [[communication theory]] for decoding [[code|codewords]] sent over a [[noisy channel]] (such as a [[binary symmetric channel]]).
==Notation==
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==Ideal observer decoding==
Given a received [[codeword]] <math>x \in \mathbb{F}_2^n</math>, '''ideal observer decoding''' picks a codeword <math>y \in C</math> to maximise:
:<math>\mathbb{P}(y \mbox{ sent} \mid x \mbox{ received})</math>
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==Maximum likelihood decoding==
Given a received codeword <math>x \in \mathbb{F}_2^n</math> '''[[maximum likelihood]] decoding''' picks a codeword <math>y \in C</math> to [[maximization|maximise]]:
:<math>\mathbb{P}(x \mbox{ received} \mid y \mbox{ sent})</math>
-the codeword that was most likely to have been sent [[conditional probability|given that]] <math>x</math> was received. Note that if all codewords are equally likely to be sent during ordinary use, then this scheme is equivalent to ''ideal observer decoding'':
:<math>
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