Syndrome Decoding/Examples/(6, 3) code in Z2

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Example of Syndrome Decoding

Let $C$ be the linear code:

$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$

Then the Syndrome Decoding table $T$ for $C$ is:

$\begin{array} {cc}

000000 & 000 \\ 100000 & 110 \\ 010000 & 101 \\ 001000 & 011 \\ 000100 & 100 \\ 000010 & 010 \\ 000001 & 001 \\ 100001 & 111 \\ \end{array}$


To find the $8$th row, it is not necessary to hunt directly an element $v$ of $\map V {6, 2}$ of weight $2$ which does not exist anywhere in the other $7$ rows.

What you do is identify the remaining syndrome of $3$ digits that has not been used yet (that is: $111$).

Then you work out what combinations of coset leaders and their own syndromes which when added together make that last syndrome.


The fact that in this case that combination is not unique (here we get $100001$, $010010$, $001100$ and $000111$) means that an element of $\map V {6, 2}$ which is more than $2$ distant from a codeword cannot be uniquely decoded.


Example

Let $C$ be the linear code:

$C = \set {000000, 100110, 010101, 110011, 001011, 101101, 011110, 111000}$

Then the Syndrome Decoding of $100111$ yields $100110$.


Sources

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