Consider the linear code

A :$=<\{(1,1),(\backslash$alpha $,1+\backslash$alpha $)\}>$

over F$4,$ where $\backslash$alpha is a root of $1+$ x $+$ x $2$ in F$2[$x$].$ Let B be the binary

code $\{0000,1100,1010,0110\}$ and consider the F$2-$linear transformation

between F$4$ and B defined by

$\backslash$phi : $0->0000,1->1100,\backslash$alpha $->1010,1+\backslash$alpha $->0110.$

Determine all the codewords of the code

C :$=\backslash$phi $($A$)=\{(\backslash$phi $($c$1),\backslash$phi $($c$2))$ : $($c$1,$ c$2)$ in A$\}$