Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword

Theorem

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

Let $\map d C$ denote the minimum distance of $C$.

Then:

$\map d C = \ds \min_{u \mathop \in C} \map w u$

where $\map w u$ denotes the weight of $u$.

Let $f := \ds \min_{u \mathop \in C} \map w u$.

Let $\mathbf 0$ denote the codeword in $\map V {n, p}$ consisting of all zeroes.

As $C$ is a subspace of $\map V {n, p}$, we have that $\mathbf 0 \in C$.

Let $w$ be a codeword with weight $f$.

Then:

$\map d {w, \mathbf 0} = f$

so $f \ge \map d C$.

Let $u, v \in C$ such that $\map d {u, v} = \map d C$.

We have that $C$ is a linear code.

Therefore:

$u - v \in C$

where $u - v$ denotes the difference between $u$ and $v$.

But $u - v$ has weight $\map d C$.

Thus:

$\map d C \ge f$

and it follows that $\map d C = f$.

$\blacksquare$

1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Proposition $6.9$