This article discusses common methods in communication theory for decoding codewords sent over a noisy channel (such as a binary symmetric channel).

Notation

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Henceforth $C\; \backslash subset\; \backslash mathbb\{F\}\_2^n$ shall be a (not necessarily linear) code of length $n$; $x,y$ shall be elements of $\backslash mathbb\{F\}\_2^n$; and $d(x,y)$ shall represent the Hamming distance between $x,y$.

Ideal observer decoding

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Given a received codeword $x\; \backslash in\; \backslash mathbb\{F\}\_2^n$, ideal observer decoding picks a codeword $y\; \backslash in\; C$ to maximise:

$\backslash mathbb\{P\}(y\; \backslash mbox\{\; sent\}\; \backslash mid\; x\; \backslash mbox\{\; received\})$

-the codeword (or a codeword) $y\; \backslash in\; C$ that is most likely to be received as $x$.

Where this decoding result is non-unique a convention must be agreed. Popular such conventions are:

1. Request that the codeword be resent;

2. Choose any one of the possible decodings at random.

Maximum likelihood decoding

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Given a received codeword $x\; \backslash in\; \backslash mathbb\{F\}\_2^n$ maximum likelihood decoding picks a codeword $y\; \backslash in\; C$ to maximise:

$\backslash mathbb\{P\}(x\; \backslash mbox\{\; received\}\; \backslash mid\; y\; \backslash mbox\{\; sent\})$

-the codeword that was most likely to have been sent given that $x$ was received. Note that if all codewords are equally likely to be sent during ordinary use, then this scheme is equivalent to ideal observer decoding:

$$

\begin{matrix} \mathbb{P}(x \mbox{ received} \mid y \mbox{ sent}) & = & \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent} )} \\ && \\ & = & \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent})} \frac{\mathbb{P}(y \mbox{ sent})}{\mathbb{P}(x \mbox{ sent})} \\ && \\ & = & \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent})} \\ && \\ & = & \mathbb{P}(y \mbox{ received} \mid x \mbox{ sent}) \\ \end{matrix}

As for ideal observer decoding, a convention must be agreed for non-unique decoding. Again, popular such conventions are:

1. Request that the codeword be resent;

2. Choose any one of the possible decodings at random.

Minimum distance decoding

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Given a received codeword $x\; \backslash in\; \backslash mathbb\{F\}\_2^n$, minimum distance decoding picks a codeword $y\; \backslash in\; C$ to minimise the Hamming distance :

$d(x,y)\; =\; \backslash \#\; \backslash \{i\; :\; x\_i\; \backslash not\; =\; y\_i\; \backslash \}$

-the codeword (or a codeword) $y\; \backslash in\; C$ that is as close as possible to $x\; \backslash in\; \backslash mathbb\{F\}\_2^n$.

Notice 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 likelhood decoding since if

$d(x,y)\; =\; d$

then:

$$

\begin{matrix} \mathbb{P}(y \mbox{ received} \mid x \mbox{ sent}) & = & (1-p)^{n-d}p^d \\ && \\ & = & (1-p)^n \left( \frac{p}{1-p}\right)^d \\ \end{matrix}

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

As for other decoding methods, a convention is agreed for non-unique decoding. Popular such conventions are:

1. Request that the codeword be resent;

2. Choose any one of the possible decodings at random.

Syndrome decoding

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Syndrome decoding is a highly efficient method of decoding a linear code over a noisy channel - ie 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.

Suppose that $C\backslash subset\; \backslash mathbb\{F\}\_2^n$ is a linear code of length $n$ and minimum distance $d$ with parity check matrix $H$. Then clearly $C$ is capable of correcting upto

$t=\backslash lfloor\backslash frac\{d-1\}\{2\}\backslash rfloor$

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 $x\; \backslash in\; \backslash mathbb\{F\}\_2^n$ is sent over the channel and the error pattern $e\; \backslash in\; \backslash mathbb\{F\}\_2^n$ 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 - ie an element (not necessarily unique) $c\; \backslash in\; C$ with

$d(c,z)\; \backslash leq\; d(y,z)$

for all $y\; \backslash in\; C$. Syndrome decoding takes advantage of the property of the parity matrix that:

$Hx\; =\; 0$

for all $x\; \backslash in\; C$. 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

$$

\begin{matrix} \sum_{i=0}^t \binom{n}{i} < |C| \\ \end{matrix}

(for a binary code) against pre-computed values of $He$ for all possible error patterns $e\; \backslash in\; \backslash mathbb\{F\}\_2^n$. Knowing what $e$ is, it is then trivial to decode $x$ as:

$x\; =\; z\; -\; e$

Notice that this will always give a unique (but not necessarily accurate) decoding result since

$Hx\; =\; Hy$

if and only if $x=y$. This is because the parity check matrix $H$ is a generator matrix for the dual code $C^\backslash perp$ and hence is of full rank.

coding theory

Méthode de décodage