Determine whether the general dual candidates form a vectorial dual of F

Determine whether the vectorial functions G\circ L_t, where G and L_t are defined from the vectorial function F(z)=\operatorname{Tr}_{q/p^k}(g_1(z))+c(1-(z+z^q)^{q-1}) and the parameter t\in\mathbb{F}_{p^k}^*, collectively constitute a vectorial dual of F when the associated permutation \pi_1 does not satisfy the additional linearized-polynomial condition imposed later in the paper.

Background

The paper studies the vectorial bent function F built from the function g_1 and examines whether its componentwise Walsh duals arise as the components of another vectorial bent function. For each nonzero t\in\mathbb{F}_{p^k}, the paper expresses the dual of the component F_t as a component of a function of the form G\circ L_t, where the linear map L_t depends on t.

The authors note that the resulting candidate G\circ L_t varies with t. Consequently, the displayed componentwise dual descriptions do not directly identify one fixed vectorial function whose components are all the duals of the components of F. The paper resolves this issue only under a special representation of \pi_1^{-1}; the general status of whether these t-dependent candidates define a vectorial dual remains unresolved in the cited remark.