A New Class of Balanced Near-Perfect Nonlinear Mappings and Its Application to Sequence Design

A mapping from Z_N to Z_M can be directly applied for the design of a sequence of period N with alphabet size M, where Z_N denotes the ring of integers modulo N. The nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence. When M is a divisor of N, the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation, but it is not balanced. In this paper, we study balanced near-perfect nonlinear (NPN) mappings applicable for the design of sequence sets with low correlation. We first construct a new class of balanced NPN mappings from Z(p²-p) to Z_p for an odd prime p. We then present a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping. By applying it to the new class, we obtain a new optimal FHS set of period p²-p with respect to the Peng-Fan bound, whose FHSs are balanced and optimal with respect to the Lempel-Greenberger bound. Moreover, we construct a low-correlation sequence set with size p, period p²-p, and maximum correlation magnitude p from the new class of balanced NPN mappings, which is asymptotically optimal with respect to the Welch bound.