Decoding methods: Difference between revisions

Revision as of 02:32, 25 February 2008

Revision as of 02:38, 25 February 2008

Line 44:

:<math>d(x,y) = \# \{i : x_i \not = y_i \}</math>

-thei.e. codewordchoose (or athe codeword) <math>y \in C</math> that is as close as possible to <math> x \in \mathbb{F}_2^n</math>.

NoticeNote that if the probability of error on a [[discrete memoryless channel]] <math>p</math> is strictly less than one half, then ''minimum distance decoding'' is equivalent to ''maximum likelihood decoding'', since if

:<math>d(x,y) = d,\,</math>

Line 54:

:<math>

\begin{align}

\mathbb{P}(y \mbox{ received} \mid x \mbox{ sent}) & {} = (1-p)^{n-d} \cdot p^d \\

& {} = (1-p)^n \cdot \left( \frac{p}{1-p}\right)^d \\

\end{align}

</math>

Line 61:

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

As forwith other decoding methods, a convention ismust be agreed to for non-unique decoding. Popular such conventions are:

:# Request that the codeword be resent;

:# Choose any one of the possible decodings at random.

==Syndrome decoding==