Condition for Vectors to have Same Syndrome

Theorem
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 $u, v \in \map V {n, p}$.

Then $u$ and $v$ have the same syndrome they are in the same coset of $C$.

Proof
Let $u, v \in \map V {n, p}$.

Let $\map S u$ denote the syndrome of $u$.

Then:

Hence the result from Elements in Same Coset iff Product with Inverse in Subgroup.