Vectorial Bent Functions With Non-Weakly Regular Components

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.