Let $P$ be a Paley matrix of order $6$ and let $G=\left[\begin{array}{c|c}I_6 & P\end{array}\right]$ be a $6\times12$ matrix. View this matrix as a ternary matrix and consider $C$ to be the ternary code generated by $G$ (over $\mathbb F_3$). Prove that $C$ is a ternary self-dual code with parameters $[12,6,6]$ and that we obtain a perfect ternary code if we puncture this code on some coordinate.
(Hint: take $q=5.$ Construct the Paley matrix of order $6$ from the conference matrix of order $5$ by using quadratic residues in $\mathbb F_5.$)