A New Class of Balanced Near-Perfect Nonlinear Mappings and Its Application to Sequence Design
A New Class of Balanced Near-Perfect Nonlinear Mappings and Its Application to Sequence Design
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DOI：
10.1109/tit.2012.2224146
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发表时间：
2013-02
影响因子：
2.5
通讯作者：
Jin-Ho Chung;Kyeongcheol Yang
中科院分区：
文献类型：
--
作者：
Jin-Ho Chung;Kyeongcheol Yang
从ZN到ZM的映射可以直接应用于具有字母表大小M的周期N的序列的设计，其中ZN表示整数模N的环。这种映射的非线性与相应序列的自相关性密切相关。当M是N的约数时，完全非线性映射所对应的序列具有完全自相关，但它是不平衡的。本文研究了平衡的近完美非线性（NPN）映射，它适用于低相关序列集的设计。首先构造了一类新的从Z（p2-p）到Zp（p为奇素数）的平衡NPN映射，然后给出了由非线性映射构造跳频序列集的一般方法.通过将其应用于新的类，我们得到了一个新的关于Peng-Fan界的周期为p2-p的最优FHS集，其FHS是平衡的，并且关于Lempel-Greenberger界是最优的。此外，我们构造了一个低相关序列集的大小为p，周期为p2-p，最大相关幅度为p的新的平衡NPN映射类，这是渐近最优的Welch界。
A mapping from ZN to ZM can be directly applied for the design of a sequence of period N with alphabet size M, where ZN 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 Zp 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.