  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 $M|q-1$, 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
$q\rightarrow \infty$, of that is $\frac{ (M-1)q^{d-1}}{d }$. This extends the
pioneering work of Yu and Gong for $d=2$ case.
