Golay Ternary Code has Minimum Distance 5

Theorem
The Golay ternary code has a minimum distance of $5$.

Proof
Let $C$ denote the Golay ternary code.

By inspection of the standard generator matrix $G$ of $C$:
 * $G := \begin{pmatrix}

1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 2 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \end{pmatrix}$

it is seen that the smallest weight of all the codewords of $C$ that can be found in $G$ is $5$.

So it is immediately seen that the minimum distance of $C$ is at least $5$.

It remains to be shown that the minimum distance of $C$ is no more than $5$.