Sequences with ideal auto-correlation derived from group actions

Bent functions have a number of practical applications in cryptography, coding theory, and other fields. Fourier transform is a key tool to study bent functions on finite abelian groups. Using Fourier transforms, in this paper, we first present two necessary and sufficient conditions on the existence of bent functions via faithful actions of finite abelian groups and then show two constructions of sequences with ideal auto-correlation (SIACs). In addition, we construct a periodic complementary sequence set (PCSS) by rearranging a periodic multiple shift sequence (PMSS) corresponding to a bent function on a finite abelian group. Some concrete constructions of SIACs and PCSSs are provided to illustrate the efficiency of our methods.