$ N$ and $ L$ ($ L<N$ ) are given constant. Suppose there are $ N$ column vectors $ a_i\in{\mathbb{C}^{L}}(i=1,2,…N)$ . The equation set for these vectors are
\begin{equation} \left\{ \begin{array}{lr} \langle a_1,a_N \rangle=0 \ \langle a_1,a_{N-1} \rangle+\langle a_2,a_{N}\rangle=0\ \langle a_1,a_{N-2} \rangle+\langle a_2,a_{N-1} \rangle+\langle a_3,a_{N}\rangle=0 \ \vdots \ \langle a_1,a_2 \rangle+\langle a_2,a_3\rangle+ \ldots \langle a_{N-1},a_N\rangle=0 \end{array} \right. \end{equation}
where $ \langle a,b \rangle$ means $ a^Hb$ . Please find $ a_i(i=1,2,…N)$ that satisfy the equation set.
