Check entry extended ternary Golay code

Question

The extended ternary Golay code is the linear $[12,6,6]$-code with the following generator matrix: $$ C=\left( \begin{array} &1&0&0&0&0&0&0&1&1&1&1&1\\ 0&1&0&0&0&0&1&0&1&2&2&1\\ 0&0&1&0&0&0&1&1&0&1&2&2\\ 0&0&0&1&0&0&1&2&1&0&1&2\\ 0&0&0&0&1&0&1&2&2&1&0&1\\ 0&0&0&0&0&1&1&1&2&2&1&0 \end{array} \right) $$ But if I look at the check-sum (the sum of the entries) of the second row, it is equal to $8$ and this is not $0\pmod 3$. What is going on? This code is supposed to be the extension of the ternary Golay code.

Answer 1

Posting my comment as an answer lest this question gets stuck in the unanswered queue.

It is a possible explanation that the intended generator matrix for the code $C$ was $$ G=\left( \begin{array} &1&0&0&0&0&0&0&1&1&1&1&1\\ 0&1&0&0&0&0&2&0&1&2&2&1\\ 0&0&1&0&0&0&2&1&0&1&2&2\\ 0&0&0&1&0&0&2&2&1&0&1&2\\ 0&0&0&0&1&0&2&2&2&1&0&1\\ 0&0&0&0&0&1&2&1&2&2&1&0 \end{array} \right) $$ It is easy to see that $GG^T=0$, and as $G$ clearly has rank six, it follows that it generates a self-dual code. So we can use $G$ as a ternary check matrix as well.

Because $1^2\equiv2^2=1\pmod3$ it follows that the Hamming weights of all the words in $C$ are multiples of three. Because no set of three columns is linearly dependent (this takes some convincing as it splits into many cases, but they all are quite straightforward), the minimum Hamming distance of $C$ is at least six. It follows from the uniqueness (up to equivalence) of the Golay codes that $C$ must be equivalent to the Golay code.

So the original code is equivalent to the ternary Golay code. The seventh symbol of every word is negated. It depends on your definititions whether that is killjoy or not.