The Structure Of Global Phase Functions On Finite Abelian Groups

In this paper, we study the structure of globally k-order phase functions for any non-negative integer k. Using the properties of global phase functions and finite abelian groups,we generalize the inverse theorem for globally quadratic phase functions [11]. In Section 2, we get the structure of globally 0-order phase functionsφ(x)=h and globally 1-order phase functions' structureφ(x)=ξ(x)+h. Moreover, on these bases, the construction of globally 3-order phase functions is obtainedWhereφ: G→H is a mapping, M_i∈S_i(G,H),h∈H,1≤i≤3. In Section 3, we furtherly obtain the structure of globally k-order phase functions for any non-negative integer kWhere M_i∈S_i(G, H), h∈H,1≤i≤k.