Equivalence for generalized Boolean functions

Equivalence plays a key-role for the classification of functions between elementary abelian groups ${{\mathbb V}}_n^{(p)}$ and ${{\mathbb V}}_k^{(p)}$. One distinguishes between affine equivalence, extended affine (EA) equivalence, and the most general CCZ-equivalence. Recently, there has been an increased interest in functions from elementary abelian groups ${{\mathbb V}}_n^{(p)}$ to cyclic groups ${{\mathbb Z}}_{p^k}$. We initiate the study of equivalence for functions from ${{\mathbb V}}_n^{(p)}$ to ${{\mathbb Z}}_{p^k}$. We show that CCZ-equivalence is more general than EA-equivalence. For some classes of functions, CCZ-equivalence reduces to EA-equivalence. We show that CCZ-equivalence between two functions from ${{\mathbb V}}_n^{(p)}$ to ${{\mathbb Z}}_{p^k}$ implies CCZ-equivalence of two associated vectorial functions from ${{\mathbb V}}_n^{(p)}$ to ${{\mathbb V}}_k^{(p)}$.