Definition:Syndrome

Definition

Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$

Let $G$ be a (standard) generator matrix for $C$.

Let $P$ be a standard parity check matrix for $C$.

Let $v \in \map V {n, p}$, considered as a vector in the vector space that is $\map V {n, p}$.

Let $v^\intercal$ denote the transpose of $v$.

The syndrome of $v$ is the element of $\map V {n - k, p}$ defined as:

$\map S v := P v^\intercal$

Examples

Linear $\tuple {6, 3}$-code in $\Z_2$

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 $000$.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Definition $6.19$