Correlation measures of binary sequences using additive characters

Mauduit, Rivat and Sárközy presented a construction of a binary sequence which utilizes properties of additive characters and polynomials, and showed that for this sequence both $W(E_N)$ and the correlations of “small” order are “small” if the order of the correlation is less than the degree of the polynomial. They conjectured that if the order of the correlation is greater than the degree of the polynomial, then the correlation is large. We further study the correlation measures of the sequence defined by Mauduit, Rivat and Sárközy and show that if the polynomial is monic with degree $d=2^{\alpha }-1$ for $\alpha \geq 2$ then the correlation of order $k=d+1=2^{\alpha }$ is very large.