SOME TERNARY SEQUENCES

Let us combine the notions of Legendre and maximal sequences choosing $\alpha$ of order $n$ in${L}^\times$and a multiplicative character of 

$$s(k)=\chi\circ{\rm tr}_{{L}/{K}}(\alpha^k),$$

    What are the correlations ?

$$\hat s(u)={nq^{s-1}(q-1)\over q^s-1}\sum_{\psi\lambda^u\chi=1}{G_K(\psi\lambda)\over G_L(\psi\lambda)}$$

where $\psi(\alpha)=1$ and $\psi\lambda^u\chi$ is trivial over $K$.

we deduce

[PL, 1994] If $n={q^s-1\overq-1}$ and $(s,q-1)=1$ the sequence  $s$ has perfect ( type I ) correlations.

[PL, 1994]

If $n=2{q^s-1\over q-1}$, $(2s,q-1)=2$, and the order of $\chi$ does not divide $q-1\over 2$ then $s$ has type II correlations.