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

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Example of Syndrome

Let $C$ be the linear $\tuple {6, 3}$-code in $\Z_2$ whose standard generator matrix $G$ is given by:

$G := \begin{pmatrix}

1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix}$


The syndrome of $100000$ is $110$

The syndrome of $110011$ is $000$.


Proof

From Standard Parity Check Matrix on the given linear $\tuple {6, 3}$-code in $\Z_2$:

$P := \begin{pmatrix}

1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{pmatrix}$

where $P$ is the standard parity check matrix of $C$.


Thus:

\(\ds P \tuple {100000}^\intercal\) \(=\) \(\ds \begin{pmatrix}

1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}\)

Definition of Syndrome
\(\ds \) \(=\) \(\ds \begin{pmatrix} 1 & 1 & 0 \end{pmatrix}\)


\(\ds P \tuple {110011}^\intercal\) \(=\) \(\ds \begin{pmatrix}

1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix}\)

Definition of Syndrome
\(\ds \) \(=\) \(\ds \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}\)

$\blacksquare$


Sources

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