A family of sequences with large size and good correlation property arising from $M$-ary Sidelnikov sequences of period $q^d-1$

Let $q$ be any prime power and let $d$ be a positive integer greater than 1. In this paper, we construct a family of $M$-ary sequences of period $q-1$ from a given $M$-ary, with $M\mathrm{\mid }q-1$, Sidelikov sequence of period $q^d-1$. Under mild restrictions on $d$, we show that the maximum correlation magnitude of the family is upper bounded by $(2d-1)\sqrt{q}+1$ and the asymptotic size, as $q\to \mathrm{\infty }$, of that is $\frac{(M-1)q^{d-1}}{d}$. This extends the pioneering work of Yu and Gong for $d=2$ case.