The Zero-Difference Properties of Functions and Their Applications
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A function f from group (A,+) onto group (B,+) is (n, m, S) zero-difference (ZD), if S is the minimal one of those sets T such that for any non-zero element a ∈ A, |{x ∈ A | f(x+a)-f(x)=0}|∈ T, where n=|A| and m=|B|. ZD is a generalization of differentially δ-vanishing, zero-difference balanced and near zero-difference balanced. In this paper, the framework of obtaining the ZD property of coset index function over algebraic rings is proposed. Then the ZD properties of coset index functions over residual class rings Z_p^k are shown, where p is a prime number and k> 2 is a positive integer. Finally, DSS-ZD functions and FHS-ZD functions are proposed in order to obtain optimal difference systems of sets (DSS) and optimal frequency-hopping sequences (FHS).
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