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=== Decoding conventions ===
Note that the probability for each codeword may not be unique: there may be more than one codeword with an equal likelihood of mutating into the received message. In such a case, the sender and receiver(s) must agree ahead of time on a decoding convention. Popular conventions include:
:# Request that the codeword be resent -- [[automatic repeat-request]]
:# Choose any random codeword from the set of most likely codewords
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==Syndrome decoding==
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'''Syndrome decoding''' is a highly efficient method of decoding a [[linear code]] over a ''noisy channel'' - ie one on which errors are made. In essence, syndrome decoding is ''minimum distance decoding'' using a reduced lookup table. It is the linearity of the code which allows for the lookup table to be reduced in size.
The simplest kind of syndrome decoding is [[Hamming code]].
Suppose that <math>C\subset \mathbb{F}_2^n</math> is a linear code of length <math>n</math> and minimum distance <math>d</math> with [[parity-check matrix]] <math>H</math>. Then clearly <math>C</math> is capable of correcting up to
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