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.

n; d(x,y) is the distance between those elements.

One may be given the message ℙ(y sent∣x received)

For example, a person can choose the codeword  – automatic repeat-request.

Choose any random codeword from the set of most likely codewords which is nearer to that.

If another code follows, mark the ambiguous bits of the codeword as erasures and hope that the outer code disambiguates them

Report a decoding failure to the system

Given a received vector ℙ(x received∣y sent),

that is, the codeword y was sent. If all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by Bayes' theorem,

x is restructured and y precisely when y∈C to minimise the Hamming distance:

x.

Note that if the probability of error on a discrete memoryless channel ℙ(y received∣x sent)=(1−p)n−d⋅pd=(1−p)n⋅(p1−p)d

which (since p is less than one half) is maximised by minimising d.

Minimum distance decoding is also known as nearest neighbour decoding. It can be assisted or automated by using a standard array. Minimum distance decoding is a reasonable decoding method when the following conditions are met:

1. The probability