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

In this paper, we extend the construction by Vu and Gong for families of M-ary sequences of period q - 1 from the array structure of an M-ary Sidelnikov sequence of period q² - 1, where q is a prime power and M|q - 1. The construction now applies to the cases of using any period q^d -1 for 3 ≤ d <; (1/2)(√q - (2/√q) + 1) and q > 27. The proposed construction results in a family of M-ary sequences of period q-1 with: 1) the correlation magnitudes, which are upper bounded by (2^{d -1})√q +1 and 2) the asymptotic size of (M -1)q^d-1/d as q increases. We also characterize some subsets of the above of size ~(r - 1)q^d-1/d but with a tighter upper bound (2d - 2)√q + 2 on its correlation magnitude. We discuss reducing both time and memory complexities for the practical implementation of such constructions in some special cases. We further give some approximate size of the newly constructed families in general and an exact count when d is a prime power or a product of two distinct primes. The main results of this paper now give more freedom of tradeoff in the design of M-ary sequence family between the family size and the correlation magnitude of the family.