I need to show that the extended ternary Golay code with parameters $[12,6]$ is a self dual code and has minimum weight $6$ so i will obtain a $[12,6,6]$-code. I showed the self duality but how can I show that the minimum weight is $6$?
Thanks for your help...
Hint: Show that the code has no words of weight three. Also use the fact that if $\vec{c}=(c_1,c_2,\ldots,c_{12})$ is a codeword, then, by self-duality, its weight $$ w=\vec{c}\cdot\vec{c}\equiv0\pmod3. $$ The first equality follows from the fact $c_i^2=1$ for all non-zero values of $c_i$.
The most convenient way of excluding the possibility of weight three words may be to use the description of the (unextended) ternary Golay code as a QR-code (=quadratic residue code), and the general results on the minimum distance of QR-codes.