Large families of pseudorandom binary sequences constructed by using the Legendre symbol (Q2889257)

A new large families of pseudorandom binary sequences \(E_{p-1}= (e_1,\dots, e_{p-1})\in \{+1,-1\}^p\) constructed from the Legendre symbol are given. Let \(p>2\) be a prime, and let \(f(z)\in\mathbb F_p[x]\) be any polynomial. The sequence \(E_{p-1}\) is defined by NEWLINE\[NEWLINEe_n= \begin{cases} x_2(f(n)+\overline n)\quad & \text{for }(f(n)+\overline n,p)= 1,\\ +1\quad & \text{for }p\mid f(n)+\overline n,\end{cases}NEWLINE\]NEWLINE where \(n\overline n\equiv 1\pmod p\), \(1\leq\overline n\leq p-1\). The pseudorandom properties (the well distribution measure and the correlation measure, etc.) of \(E_{p-1}\) are studied by using an estimate for character sums.