There is no $0$-perfect sequences

 
Let $s$ be a sequence of period $v=2\ell$ satisfying
\begin{displaymath}s_{i+\ell} = -s_{i},\quad\forall i\end{displaymath}


If $s\times s(1)=0$ then $\ell$ is odd.
 Proof :

\begin{displaymath}\begin{split}s\times s(1) &=0\\&=\sum_{i=0}^{v-1} s_i\,s_{i+1}\\&=2 \sum_{i=0}^{\ell-1} s_i\,s_{i+1}\\\end{split}\end{displaymath}


In the sum
\begin{displaymath}s_0s_1+s_1s_2+\dots +s_{\ell-1}s_\ell\end{displaymath}
we replace $s_\ell$ by $-s_0$. Each $s_i$ appears two times, because of the sign the products of all the terms is equal to $-1$. On the other hand, the number of terms equal to $-1$ is $\ell\over 2$...