Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation $f$ (not) belong to the completed Maiorana-McFarland class $\mathcal{M}^\#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-subspaces for the concatenation of the form $f=f_1||f_2$ and $f=f_1||f_2||f_3||f_4$, which allows us to specify the necessary and sufficient conditions so that $f$ is outside $\mathcal{M}^\#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^\#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.

翻译：每个布尔弯曲函数$f$要么可以写成两个互补半弯曲函数$f_1,f_2$的拼接$f=f_1||f_2$，要么可以写成四个布尔函数$f_1,f_2,f_3,f_4$的拼接$f=f_1||f_2||f_3||f_4$，且这四个函数同时属于弯曲函数、半弯曲函数或5值谱函数。在此背景下，一个关键问题自然产生：弯曲拼接$f$何时（不）属于完成的Maiorana-McFarland类$\mathcal{M}^\#$？本文通过完整刻画形如$f=f_1||f_2$与$f=f_1||f_2||f_3||f_4$的拼接中$\mathcal{M}$-子空间的结构，完全回答了该问题，从而给出了$f$不属于$\mathcal{M}^\#$的充要条件。基于这些条件，我们针对$f=g||h||g||(h+1)$（其中$g$和$h$均为弯曲函数）的特殊情形，提出了若干显式构造不属于$\mathcal{M}^\#$的弯曲函数的设计方法。