Let $C$ be a binary code and let $\overline{C}$ be the extended code of $C$ obtained by adding an overall parity check.
Is it true that
$W_{\overline{C}}(x)=\frac{1}{2}[(1+x)W_C (x)+(1-x)W_{C}(-x)]$
Where $W_{C}$ is the weight enumerator $\displaystyle W_C= \sum_{i=0}^{n} A_{i}(C)x^i$.
I can't find a counterexample about this relation so I assume that it must be true but I have no idea how to show this. Any hint and help would be much appreciated!!