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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. 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. C ⊂ F 2 n {displaystyle Csubset mathbb {F} _{2}^{n}} is considered a binary code with the length n {displaystyle n} ; x , y {displaystyle x,y} shall be elements of F 2 n {displaystyle mathbb {F} _{2}^{n}} ; and d ( x , y ) {displaystyle d(x,y)} is the distance between those elements. One may be given the message x ∈ F 2 n {displaystyle xin mathbb {F} _{2}^{n}} , then ideal observer decoding generates the codeword y ∈ C {displaystyle yin C} . The process results in this solution: For example, a person can choose the codeword y {displaystyle y} that is most likely to be received as the message x {displaystyle x} after transmission.

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