I am looking for a direct proof of this statement:
"A $12\times 12$ Hadamard matrix is the generator matrix of a ternary selfdual (linear) $[12,6,6]$ code $C$."
That the length is 12 is clear. As the rows of a Hadamard matrix are orthogonal, the code $C$ is self-orthogonal ($C\subseteq C^\bot$). The code has minimum weight at most 6 also follows directly (if the first row of the Hadamard matrix is made into all 1's, subtracting the second row which has equally many 1's as -1's yields a code word of weight 6).
However, if fail to see (1) that a Hadamard 12x12-matrix has rank 6, or that that $C^\bot\subseteq C$ and (2) why the minimum weight cannot be smaller than 6.
The reason why I ask for a 'direct' proof is that I suspect these codes might have to do something with the ternary Golay code which I want to avoid using...