An important classification of permutations over \(\mathbb {F}_2^m\) , suitable for constructing Maiorana-McFarland bent functions on \(\mathbb {F}_2^m \times \mathbb {F}_2^m\) with the unique \(\mathcal {M}\) -subspace of maximal dimension, was recently considered in Pasalic et al (IEEE Trans Inf Theory 70:4464–4477, 2024). More precisely, two properties called \((P_1)\) and \((P_2)\) were introduced and a generic method of constructing permutations having the property \((P_1)\) was presented, whereas no such results were provided related to the \((P_2)\) property. In this article, we provide a deeper insight on these properties, their mutual relationship, and specify some explicit classes of permutations having these properties. Such permutations are then employed to generate a large variety of bent functions outside the completed Maiorana-McFarland class \({\mathcal {M}}^\#\) . We also introduce \(\ell \) -optimal bent functions as bent functions with the lowest possible linearity index; such functions can be considered as opposite to Maiorana-McFarland bent functions. We give explicit constructions of \(\ell \) -optimal bent functions within the \({{\mathcal {D}}}_0\) class, which in turn can be employed in certain secondary constructions of bent functions (Zhang et al in Inf Comput 297:105149, 2024) for providing even more classes of bent functions that are provably outside \({\mathcal {M}}^\#\) . Moreover, we demonstrate that a certain subclass of \({{\mathcal {D}}}_0\) has an additional property of having only 5-valued spectra decompositions, similarly to the only result in this direction concerning monomial bent functions (Canteaut and Charpin in IEEE Trans Inf Theory 498:2004–2019, 2003). Finally, we generalize the so-called swapping variables method introduced in Pasalic et al. (IEEE Trans Inf Theory 70:4464–4477, 2024) which then allows us to specify much larger families of bent functions outside \({\mathcal {M}}^\#\) compared to Pasalic et al (IEEE Trans Inf Theory 70:4464–4477, 2024). In this way, we give a better explanation of the origin of bent functions in dimension eight, since the vast majority of them is outside \(\mathcal {M}^\#\) , as indicated in Langevin and Leander (Designs Codes Cryptogr 59:193–205, 2011).
