Construction and equivalence for generalized boolean functions

Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun., 18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from $V_n^{\left(p\right)}$ to the cyclic group $Z_{p^k}$ has been initiated, where $V_n^{\left(p\right)}$ denotes an n-dimensional vector space over $F_p$ . Amongst others it has been shown that there exist functions from $V_n^{\left(2\right)}$ to $Z_4$ which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from $V_n^{\left(p\right)}$ to $Z_{p^k}$ . We then discuss constructions of generalized bent functions from $V_n^{\left(p\right)}$ to $Z_{p^k}$ , p odd or $p=2$ and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory 69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from $V_n^{\left(2\right)}$ to $Z_{2^k}$ , n odd, from arbitrary generalized bent functions from $V_{n-1}^{\left(2\right)}$ to $Z_{2^{k-1}}$ .