New Families of $M$-Ary Sequences With Low Correlation Constructed From Sidel'nikov Sequences

In this correspondence, for a positive integer
M
and a prime p such that M|p
ⁿ
- 1, three families of M-ary sequences using the
M
-ary Sidel'nikov sequences with period p
ⁿ
- 1 are constructed. Two small families contain [(p
ⁿ
- 1)/2]+
M
-2 or p
ⁿ
+
M
-3
M
-ary sequences, and both of their maximum magnitudes of correlation values are upper bounded by 2 radic{p
ⁿ
} + 6. The largest family has its maximum magnitude of correlation values upper bounded by 3 radic{p
ⁿ
}+5 and the family size is (
M
-1)
²
(2
^n-1
-1) +
M
-1 for p = 2 or (
M
-1)
²
(p
ⁿ
- 3)/2 +
M
(
M
-1)/2 for an odd prime p.