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.
Notation
is considered a binary code with the length ; shall be elements of ; and is the distance between those elements.
Ideal observer decoding
One may be given the message , then ideal observer decoding generates the codeword . The process results in this solution:
For example, a person can choose the codeword that is most likely to be received as the message after transmission.
Decoding conventions
Each codeword does not have an expected possibility: 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:
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is considered a 
; 
shall be elements of 
; and 
is the distance between those elements.
, then ideal observer decoding generates the codeword 
. The process results in this solution:
that is most likely to be received as the message 
after transmission.