Decoding methods

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 C\subset \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 x\in \mathbb {F} _{2}^{n}} , then ideal observer decoding generates the codeword y ∈ C {\displaystyle y\in 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. 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: Given a received vector x ∈ F 2 n {\displaystyle x\in \mathbb {F} _{2}^{n}} maximum likelihood decoding picks a codeword y ∈ C {\displaystyle y\in C} that maximizes that is, the codeword y {\displaystyle y} that maximizes the probability that x {\displaystyle x} was received, given that y {\displaystyle y} was sent. If all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding.