 Decoding methods

In communication theory and coding theory, decoding is the process of translating received messages into codewords of a given code. There has 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.
Contents
Notation
Henceforth, could have been considered a code with the length n; x,y shall be elements of ; and d(x,y) would be representing the Hamming distance between x,y. Note that C is not necessarily linear.
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 y that is most likely to be received as the message x after transmission.
Decoding conventions
Each codeword does not have a 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:

 Request that the codeword be resent  automatic repeatrequest
 Choose any random codeword from the set of most likely codewords which is nearer to that.
Maximum likelihood decoding
Further information: Maximum likelihoodGiven a received codeword maximum likelihood decoding picks a codeword to maximize:
i.e. choose the codeword y that maximizes the probability that x was received, given that y was sent. Note that if all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by Bayes Theorem we have
Upon fixing , x is restructured and is constant as all codewords are equally likely to be sent. Therefore is maximised as a function of the variable y precisely when is maximised, and the claim follows.
As with ideal observer decoding, a convention must be agreed to for nonunique decoding.
The ML decoding problem can also be modeled as an integer programming problem.^{[1]}
Minimum distance decoding
Given a received codeword , minimum distance decoding picks a codeword to minimise the Hamming distance :
i.e. choose the codeword y that is as close as possible to x.
Note that if the probability of error on a discrete memoryless channel p is strictly less than one half, then minimum distance decoding is equivalent to maximum likelihood decoding, since if
then:
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:

 The probability p that an error occurs is independent of the position of the symbol
 Errors are independent events  an error at one position in the message does not affect other positions
These assumptions may be reasonable for transmissions over a binary symmetric channel. They may be unreasonable for other media, such as a DVD, where a single scratch on the disk can cause an error in many neighbouring symbols or codewords.
As with other decoding methods, a convention must be agreed to for nonunique decoding.
Syndrome decoding
Syndrome decoding is a highly efficient method of decoding a linear code over a noisy channel  i.e. 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 is a linear code of length n and minimum distance d with paritycheck matrix H. Then clearly C is capable of correcting up to
errors made by the channel (since if no more than t errors are made then minimum distance decoding will still correctly decode the incorrectly transmitted codeword).
Now suppose that a codeword is sent over the channel and the error pattern occurs. Then z = x + e is received. Ordinary minimum distance decoding would lookup the vector z in a table of size  C  for the nearest match  i.e. an element (not necessarily unique) with
for all . Syndrome decoding takes advantage of the property of the parity matrix that:
 Hx = 0
for all . The syndrome of the received z = x + e is defined to be:
 Hz = H(x + e) = Hx + He = 0 + He = He
Under the assumption that no more than t errors were made during transmission, the receiver looks up the value He in a table of size
(for a binary code) against precomputed values of He for all possible error patterns . Knowing what e is, it is then trivial to decode x as:
 x = z − e
Partial response maximum likelihood
Main article: PRMLPartial response maximum likelihood (PRML) is a method for converting the weak analog signal from the head of a magnetic disk or tape drive into a digital signal.
Viterbi decoder
Main article: Viterbi decoderA Viterbi decoder uses the viterbi algorithm for decoding a bitstream that has been encoded using forward error correction based on a convolutional code. The Hamming distance is used as a metric for hard decision viterbi decoders. The squared Euclidean distance is used as a metric for soft decision decoders.
See also
References
 ^ "Using linear programming to Decode Binary linear codes," J.Feldman, M.J.Wainwright and D.R.Karger, IEEE Transactions on Information Theory, 51:954972, March 2005.
 Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. ISBN 0198538030.
 Pless, Vera (1982). Introduction to the theory of errorcorrecting codes. WileyInterscience Series in Discrete Mathematics. John Wiley & Sons. ISBN 0471086843.
 J.H. van Lint (1992). Introduction to Coding Theory. GTM. 86 (2nd ed ed.). SpringerVerlag. ISBN 3540548947.
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