We define a class of productive $\sigma$-ideals of subsets of the Cantor space 2$^\omega$ and observe that both $\sigma$-ideals of meagre sets and of null sets are in this class. From every productive $\sigma$-ideal $\mathscr{I}$ we produce a $\sigma$-ideal $\mathscr{I}_\kappa$, of subsets of the generalized Cantor space 2$^\kappa$. In particular, starting from meagre sets and null sets in 2$^\omega$ we obtain meagre sets and null sets in 2$^\kappa$, respectively. Then we investigate additivity, covering number, uniformity and cofinality of $\mathscr{I}_\kappa$. For example, we show that $\text{non}(\mathscr{I} = \text{non}(\mathscr{I}_{\omega_1}) = \text{non}(\mathscr{I}_{\omega_2}).$ Our results generalizes those from [5].