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 $5$.

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Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $6$: Error-correcting codes: Example $6.15$