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:


 * $\map S v := P v^\intercal$