Write F$4=\{0,1,\backslash$alpha $,\backslash$alpha $2\}($cf$.$ Example $3\mathrm{.}3\mathrm{.}5).$ Show that the following

linear codes over F$4$ are self$-$dual with respect to the Hermitian inner

product:

$($a$)$ C$1=\{(0,0),(1,1),(\backslash$alpha $,\backslash$alpha $),(\backslash$alpha $2,\backslash$alpha $2)\}$;

$($b$)$ C$2$ is the F$4-$linear code with generator matrix

$$

$1001\backslash$alpha $\backslash$alpha

$010\backslash$alpha $1\backslash$alpha

$001\backslash$alpha $\backslash$alpha $1.$

$($Note: the code C$2$ is called the hexacode.$)$

Are C$1$ and C$2$ self$-$dual with respect to the Euclidean inner product?