It is shown that the generalized Rothaus construction of p-ary bent functions can be extended to a construction of a vectorial bent function with non-weakly regular components, for which in general the duals are not a bent function, i.e., they belong to the class of non-dual bent functions. This complements results on other two constructions of non-weakly regular bent functions, the generalized Maiorana-McFarland construction and the semi-direct sum, for which vectorial versions are presented and the properties of their duals are investigated in the literature. The distribution of the values of the Walsh transform of vectorial bent functions (with non-weakly regular components) is then analysed in detail. Among others, a condition on the values of the Walsh transform of a vectorial bent function from$\mathbb {F}_{p}^{n}$to$\mathbb {F}_{p}^{m}$is presented, which implies that$m \le \lceil n/2\rceil $. This refines a classical result by Nyberg, which states that for an$(n,m)$bent function, n even, with only regular components, m can be at most$n/2$. Some results on the weight distribution of codes obtained from vectorial bent functions with non-weakly regular components complement the article.

Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs, since their duals are also bent functions. In general, this does not apply to non-weakly regular bent functions. However, the first known construction of non-weakly regular bent functions by Çeşmelioğlu et al. yields bent functions for which the dual is also bent. In this paper, the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Çeşmelioğlu et al., one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.

We show that bent functions $f$ from ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$ to ${\mathbb F}_p$, which are constant or affine on the elements of a given spread of ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$, either arise from partial spread bent functions, or they are Boolean and a generalization of Dillon's class $H$. For spreads of a presemifield $S$, we show that a bent function of the second class corresponds to an o-polynomial of a presemifield in the Knuth orbit of $S$. In contrast to the finite fields case, we have to consider pairs of (pre)semifields in a Knuth orbit. We give a canonical example of an o-polynomial for commutative presemifields (which also defines a hyperoval on the semifield plane) and show that the corresponding bent functions belong to the completed Maiorana--McFarland class. Using Albert's twisted fields and Kantor's family of presemifields, we explicitly present examples of such bent functions.

In this paper, a technique for constructing$p$-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from near-bent functions. The Fourier spectrum of quadratic monomials is analyzed, and examples of quadratic functions with highest possible absolute values in their Fourier spectrum are given. Applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when$p=$3,5. Until now, no construction of bent functions attaining these bounds was known.