New Construction of $M$-Ary Sequence Families With Low Correlation From the Structure of Sidelnikov Sequences

For prime
p
and a positive integer
m
, it is shown that
M
-ary Sidelnikov sequences of period
p
^2m
-1, if
M
|
p
^m
-1, can be equivalently generated by the operation of elements in a finite field
GF(p
^m
), including a
p
^m
-ary
m
-sequence. From the
(p
^m
-1) ×(
p
^m
+1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct new
M
-ary sequence families of period
p
^m
-1. In particular, new
M
-ary sequence families of period
p
^m
-1 are constructed from the combination of the column sequence families and known Sidelnikov-based sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the new
M
-ary sequence family of period
p
^m
-1 and the maximum correlation magnitude
2√{p
^m
}+6 asymptotically achieves
√2
times the equality of the Sidelnikov's lower bound when
M
=
p
^m
-1 for odd prime
p
.