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 Vₙ⁽ᵖ⁾ V n ( p ) to the cyclic group Zpᵏ Z p k has been initiated, where Vₙ⁽ᵖ⁾ V n ( p ) denotes an n -dimensional vector space over Fₚ F p . Amongst others it has been shown that there exist functions from Vₙ⁽²⁾ V n ( 2 ) to Z₄ Z 4 which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from Vₙ⁽ᵖ⁾ V n ( p ) to Zpᵏ Z p k . We then discuss constructions of generalized bent functions from Vₙ⁽ᵖ⁾ V n ( p ) to Zpᵏ Z p k , p odd or $$p=2$$ 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 Vₙ⁽²⁾ V n ( 2 ) to Z2ᵏ Z 2 k , n odd, from arbitrary generalized bent functions from Vₙ₋₁⁽²⁾ V n - 1 ( 2 ) to Z_2ᵏ⁻¹ Z 2 k - 1 .