Vectorial Dual-Bent Functions Overview

• Vectorial dual-bent functions are mappings where every nonzero linear form yields a bent function and the scalar duals coherently structure an m-dimensional Fₚ-vector space.
• Structural conditions like Condition A and homogeneity regimes ensure regularity or weak regularity, facilitating secondary constructions and controlled cross-correlation in codebooks.
• Explicit p-ary and Boolean concatenation methods link these functions to combinatorial designs, Gaussian sums, and optimal linear code constructions.

Vectorial dual-bent functions are vectorial bent mappings whose scalar component duals close into a coherent vectorial structure. In the standard $p$-ary formulation, a vectorial function $F: V_n^{(p)} \to V_m^{(p)}$ is vectorial bent when every nonzero linear form $u\cdot F$ is bent, and it is vectorial dual-bent when the family of duals $\{(u\cdot F)^*:u\in V_m^{(p)}\}$, together with the zero function, forms an $m$-dimensional $\mathbb{F}_p$-vector space of bent functions; equivalently, there exist a vectorial bent $G$ and a permutation $o$ such that $(u\cdot F)^*=o(u)\cdot G$ for all $u$ (Wang et al., 2023). In the weakly regular setting one may assemble the scalar duals into a unique vectorial mapping $F: V_n^{(p)} \to V_m^{(p)}$0 characterized by $F: V_n^{(p)} \to V_m^{(p)}$1 (Bapić, 2022). A closely related Boolean viewpoint organizes designated component quadruples so that duals satisfy affine identities such as $F: V_n^{(p)} \to V_m^{(p)}$2, which provides a binary, componentwise realization of dual-bent structure (Polujan et al., 2023).

1. Formal framework

For a scalar $F: V_n^{(p)} \to V_m^{(p)}$3-ary function $F: V_n^{(p)} \to V_m^{(p)}$4, the generalized Walsh–Hadamard transform is

$F: V_n^{(p)} \to V_m^{(p)}$5

The function is bent when $F: V_n^{(p)} \to V_m^{(p)}$6 for all $F: V_n^{(p)} \to V_m^{(p)}$7. In the weakly regular case,

$F: V_n^{(p)} \to V_m^{(p)}$8

with $F: V_n^{(p)} \to V_m^{(p)}$9 independent of $u\cdot F$0, and $u\cdot F$1 is the dual bent function. For vectorial $u\cdot F$2, each nonzero $u\cdot F$3 defines a scalar component $u\cdot F$4, and vectorial bentness requires every such component to be bent (Wang et al., 2023).

Two equivalent descriptions of vectorial dual-bentness are standard in the recent literature. One states that there exist a vectorial bent $u\cdot F$5 and a permutation $u\cdot F$6 on the nonzero output coefficients such that

$u\cdot F$7

for all $u\cdot F$8. Another, used prominently in odd-characteristic work, is that for $u\cdot F$9, there exists a vectorial bent $\{(u\cdot F)^*:u\in V_m^{(p)}\}$0 and a permutation $\{(u\cdot F)^*:u\in V_m^{(p)}\}$1 of $\{(u\cdot F)^*:u\in V_m^{(p)}\}$2 such that the dual of

$\{(u\cdot F)^*:u\in V_m^{(p)}\}$3

equals $\{(u\cdot F)^*:u\in V_m^{(p)}\}$4 for all $\{(u\cdot F)^*:u\in V_m^{(p)}\}$5 (Wang et al., 2022). Proposition 1 of (Wang et al., 2023) further states that for a fixed permutation $\{(u\cdot F)^*:u\in V_m^{(p)}\}$6, the corresponding vectorial dual $\{(u\cdot F)^*:u\in V_m^{(p)}\}$7 is unique.

In the Boolean concatenation setting, the dual notion is componentwise rather than field-trace based. If $\{(u\cdot F)^*:u\in V_m^{(p)}\}$8 is organized from bent Boolean components, the “vectorial dual” is understood as $\{(u\cdot F)^*:u\in V_m^{(p)}\}$9, and prescribed affine relations among the component duals define the relevant dual-bent behavior. The canonical example is the four-concatenation criterion: the concatenation $m$0 is bent if and only if

$m$1

(Polujan et al., 2023).

2. Dual alignment, homogeneity, and structural regimes

A central refinement is “Condition A,” introduced for even $m$2 and $m$3. A vectorial dual-bent $m$4 satisfies Condition A when

$m$5

for all $m$6, and all nonzero components $m$7 are either regular or weakly regular with the same constant $m$8. For $m$9, Condition A reduces to the identity alignment above; for odd $\mathbb{F}_p$0, Proposition 2 states that Condition A forces regularity, so $\mathbb{F}_p$1 (Wang et al., 2023).

A broader homogeneity regime appears in the codebook and code constructions. Under Condition I in (Heng et al., 29 Jun 2025), one assumes

$\mathbb{F}_p$2

with all components weakly regular and sharing the same sign $\mathbb{F}_p$3. Condition II keeps the same dual-scaling law and homogeneity but allows the component signs to vary as $\mathbb{F}_p$4, where $\mathbb{F}_p$5 is the quadratic character on $\mathbb{F}_p$6. These two regimes control both exact hybrid character sums and the resulting cross-correlation spectra of codebooks (Heng et al., 29 Jun 2025).

The scalar $\mathbb{F}_p$7-form viewpoint furnishes another structural characterization. A $\mathbb{F}_p$8-ary function $\mathbb{F}_p$9 is an $G$0-form if $G$1 for all $G$2. In the formulation summarized in (Wang et al., 2023), a $G$3-ary function $G$4 with $G$5 is weakly regular vectorial dual-bent if and only if $G$6 is weakly regular bent of $G$7-form with $G$8. This statement explains why homogeneity repeatedly appears in dual-scaling laws and in constructions derived from scalar weakly regular bent functions (Wang et al., 2023).

Maiorana–McFarland families provide the most explicit large classes. In the $G$9-ary setting, if $o$0 and

$o$1

then the scalar components have explicit duals involving $o$2, and these duals can be made to satisfy the $o$3 property used in secondary constructions. In the Boolean setting, the same class underlies the four-concatenation theory via functions of the form $o$4 and the $o$5 property for the permutations $o$6 (Bapić, 2022, Polujan et al., 2023).

3. Construction methods

One major secondary construction begins with a weakly regular vectorial bent $o$7 whose dual components satisfy the $o$8 property. For a linearly independent set $o$9, the $(u\cdot F)^*=o(u)\cdot G$0-ary $(u\cdot F)^*=o(u)\cdot G$1 condition is

$(u\cdot F)^*=o(u)\cdot G$2

and Lemma 1 states that this is equivalent to

$(u\cdot F)^*=o(u)\cdot G$3

for all $(u\cdot F)^*=o(u)\cdot G$4. If $(u\cdot F)^*=o(u)\cdot G$5, then the constructed function

$(u\cdot F)^*=o(u)\cdot G$6

is again vectorial weakly regular bent, and its scalar duals satisfy

$(u\cdot F)^*=o(u)\cdot G$7

(Bapić, 2022). This yields infinite families, especially in $(u\cdot F)^*=o(u)\cdot G$8-ary Maiorana–McFarland form.

A second construction paradigm is Boolean four-concatenation. If $(u\cdot F)^*=o(u)\cdot G$9 are Boolean bent functions, their canonical concatenation is bent exactly when $u$0. The paper (Polujan et al., 2023) generalizes the known Maiorana–McFarland case $u$1 to

$u$2

where $u$3 is arbitrary, and derives an explicit dual condition in terms of the inverses of permutations with the $u$4 property. The same paper also gives a piecewise lifting from $u$5-permutations to $u$6-permutations, thereby producing a recursive dimension-lifting method for new bent families satisfying the dual bent condition.

The Boolean construction is not restricted to completed Maiorana–McFarland equivalence. Under additional hypotheses involving property $u$7, the absence of linear structures in the components of $u$8, and $u$9, the resulting concatenation is bent and outside $F: V_n^{(p)} \to V_m^{(p)}$00. The same framework yields explicit monomial-permutation constructions and is further adapted to homogeneous cubic bent functions, an area where, as the paper notes, only very few design methods are known (Polujan et al., 2023).

4. Combinatorial correspondences

Vectorial dual-bent functions organize several classical combinatorial objects through their fibers. For $F: V_n^{(p)} \to V_m^{(p)}$01, let $F: V_n^{(p)} \to V_m^{(p)}$02. Under the standing assumptions used in the partial-difference-set theory—$F: V_n^{(p)} \to V_m^{(p)}$03 odd, $F: V_n^{(p)} \to V_m^{(p)}$04, $F: V_n^{(p)} \to V_m^{(p)}$05, and all nonzero components regular or weakly regular with a common sign $F: V_n^{(p)} \to V_m^{(p)}$06—the zero fiber $F: V_n^{(p)} \to V_m^{(p)}$07 is a partial difference set, and so are the preimages of squares, nonsquares, and, more generally, the preimages of cosets $F: V_n^{(p)} \to V_m^{(p)}$08 of power subgroups whenever the permutation $F: V_n^{(p)} \to V_m^{(p)}$09 is compatible with the coset structure (Wang et al., 2022). The same paper shows that if $F: V_n^{(p)} \to V_m^{(p)}$10 is the identity then the preimage of any subset $F: V_n^{(p)} \to V_m^{(p)}$11 is a partial difference set, and it interprets many earlier constructions from weakly regular $F: V_n^{(p)} \to V_m^{(p)}$12-ary bent functions as special cases of this vectorial dual-bent framework.

Under Condition A, the correspondence becomes sharper. Proposition 3 of (Wang et al., 2023) states that $F: V_n^{(p)} \to V_m^{(p)}$13 is vectorial dual-bent with Condition A if and only if every fiber $F: V_n^{(p)} \to V_m^{(p)}$14 is a regular partial difference set with explicit parameters. Theorem 1 then upgrades these fibers to an amorphic association scheme on $F: V_n^{(p)} \to V_m^{(p)}$15, with $F: V_n^{(p)} \to V_m^{(p)}$16 except in the exceptional case $F: V_n^{(p)} \to V_m^{(p)}$17, $F: V_n^{(p)} \to V_m^{(p)}$18, $F: V_n^{(p)} \to V_m^{(p)}$19, where $F: V_n^{(p)} \to V_m^{(p)}$20. The same paper further characterizes Condition A through two-weight projective codes and generalized Hadamard matrices, and in the binary case it proves that Condition A is equivalent to a bent partition structure (Wang et al., 2023).

Bent partitions provide the converse language in odd characteristic. For $F: V_n^{(p)} \to V_m^{(p)}$21 odd, Theorem 3 of (Wang et al., 2023) states that a partition $F: V_n^{(p)} \to V_m^{(p)}$22 of $F: V_n^{(p)} \to V_m^{(p)}$23 satisfies Condition $F: V_n^{(p)} \to V_m^{(p)}$24 if and only if the associated map $F: V_n^{(p)} \to V_m^{(p)}$25 for $F: V_n^{(p)} \to V_m^{(p)}$26 is a vectorial dual-bent function satisfying Condition A. This correspondence yields an alternative proof that the families $F: V_n^{(p)} \to V_m^{(p)}$27, obtained from suitable (pre)semifields, are bent partitions. It also gives a sufficient condition for unions of partition blocks to form partial difference sets and, for odd $F: V_n^{(p)} \to V_m^{(p)}$28, a characterization of such bent partitions entirely in terms of those partial difference sets (Wang et al., 2023).

5. Character sums, codebooks, and linear codes

Hybrid character sums expose the harmonic-analytic content of vectorial dual-bentness. For a multiplicative character $F: V_n^{(p)} \to V_m^{(p)}$29 of $F: V_n^{(p)} \to V_m^{(p)}$30, a vectorial dual-bent $F: V_n^{(p)} \to V_m^{(p)}$31, and $F: V_n^{(p)} \to V_m^{(p)}$32, consider

$F: V_n^{(p)} \to V_m^{(p)}$33

Under Condition I and $F: V_n^{(p)} \to V_m^{(p)}$34, Theorem 111 of (Heng et al., 29 Jun 2025) gives

$F: V_n^{(p)} \to V_m^{(p)}$35

so $F: V_n^{(p)} \to V_m^{(p)}$36. For a general nontrivial $F: V_n^{(p)} \to V_m^{(p)}$37 of order $F: V_n^{(p)} \to V_m^{(p)}$38, the same paper expresses $F: V_n^{(p)} \to V_m^{(p)}$39 through Gaussian sums and a character $F: V_n^{(p)} \to V_m^{(p)}$40 satisfying $F: V_n^{(p)} \to V_m^{(p)}$41, and again obtains modulus $F: V_n^{(p)} \to V_m^{(p)}$42 or $F: V_n^{(p)} \to V_m^{(p)}$43. Under Condition II the quadratic-character case acquires an explicit $F: V_n^{(p)} \to V_m^{(p)}$44-scaled Gaussian factor, while the nonquadratic case still yields modulus $F: V_n^{(p)} \to V_m^{(p)}$45 or $F: V_n^{(p)} \to V_m^{(p)}$46 (Heng et al., 29 Jun 2025).

These estimates feed directly into codebook constructions. The paper (Heng et al., 29 Jun 2025) derives three families of asymptotically optimal complex codebooks with very small alphabet sizes, and all three families have only two-valued or three-valued cross-correlation amplitudes. The later paper “New families of asymptotically optimal codebooks from vectorial dual-bent functions” constructs several further families that asymptotically achieve the Welch bound; the maximum cross-correlation amplitudes and the distributions of the cross-correlation amplitudes are explicitly determined, the parameter sets are new, and some families have very small alphabet sizes (Wei et al., 29 Jun 2026).

Vectorial dual-bent functions also generate linear codes through trace evaluation on fiber sets. In (Wang et al., 2024), the code

$F: V_n^{(p)} \to V_m^{(p)}$47

is constructed from a vectorial dual-bent $F: V_n^{(p)} \to V_m^{(p)}$48 and a subset $F: V_n^{(p)} \to V_m^{(p)}$49 of the codomain. Under Conditions I–III of that paper, the resulting $F: V_n^{(p)} \to V_m^{(p)}$50-ary codes are self-orthogonal; in various cases they have at most five or six nonzero weights; and their duals furnish at least almost optimal linear complementary dual codes and quantum codes. The same work shows that earlier constructions from weakly regular $F: V_n^{(p)} \to V_m^{(p)}$51-ary bent functions and from non-degenerate quadratic forms are recovered as special cases (Wang et al., 2024).

6. Explicit families, constraints, and open directions

Several concrete families illustrate the range of the theory. In the $F: V_n^{(p)} \to V_m^{(p)}$52-ary secondary construction, Example 1 of (Bapić, 2022) takes $F: V_n^{(p)} \to V_m^{(p)}$53, $F: V_n^{(p)} \to V_m^{(p)}$54, $F: V_n^{(p)} \to V_m^{(p)}$55, $F: V_n^{(p)} \to V_m^{(p)}$56, $F: V_n^{(p)} \to V_m^{(p)}$57, and $F: V_n^{(p)} \to V_m^{(p)}$58, producing a weakly regular bent $F: V_n^{(p)} \to V_m^{(p)}$59-function with explicit component duals. In the Boolean concatenation setting, (Polujan et al., 2023) gives an example with $F: V_n^{(p)} \to V_m^{(p)}$60, $F: V_n^{(p)} \to V_m^{(p)}$61, monomial permutations $F: V_n^{(p)} \to V_m^{(p)}$62, and affine trace terms $F: V_n^{(p)} \to V_m^{(p)}$63 for which the concatenation $F: V_n^{(p)} \to V_m^{(p)}$64 is bent and outside $F: V_n^{(p)} \to V_m^{(p)}$65. The same paper also provides a quadratic inverse-monomial example in $F: V_n^{(p)} \to V_m^{(p)}$66 whose concatenation is computationally verified to lie outside $F: V_n^{(p)} \to V_m^{(p)}$67 (Polujan et al., 2023).

The combinatorial side also has explicit small examples. For $F: V_n^{(p)} \to V_m^{(p)}$68 and $F: V_n^{(p)} \to V_m^{(p)}$69, the ternary weakly regular bent $F: V_n^{(p)} \to V_m^{(p)}$70-form $F: V_n^{(p)} \to V_m^{(p)}$71 has level-set sizes $F: V_n^{(p)} \to V_m^{(p)}$72, and the induced bent partition does not come from a normal bent partition, answering an open problem posed in earlier bent-partition work (Wang et al., 2023). On the coding side, (Wang et al., 2024) gives, for $F: V_n^{(p)} \to V_m^{(p)}$73, $F: V_n^{(p)} \to V_m^{(p)}$74, $F: V_n^{(p)} \to V_m^{(p)}$75, $F: V_n^{(p)} \to V_m^{(p)}$76, the self-orthogonal $F: V_n^{(p)} \to V_m^{(p)}$77 code from $F: V_n^{(p)} \to V_m^{(p)}$78 with $F: V_n^{(p)} \to V_m^{(p)}$79, and the $F: V_n^{(p)} \to V_m^{(p)}$80 code with $F: V_n^{(p)} \to V_m^{(p)}$81. For codebooks, (Heng et al., 29 Jun 2025) presents a Condition I example with $F: V_n^{(p)} \to V_m^{(p)}$82, $F: V_n^{(p)} \to V_m^{(p)}$83, $F: V_n^{(p)} \to V_m^{(p)}$84, $F: V_n^{(p)} \to V_m^{(p)}$85, yielding a $F: V_n^{(p)} \to V_m^{(p)}$86 codebook with $F: V_n^{(p)} \to V_m^{(p)}$87 (Heng et al., 29 Jun 2025).

The constraints appearing across these papers are rigid and structural. Bentness imposes even $F: V_n^{(p)} \to V_m^{(p)}$88 in the Boolean case; many vectorial constructions assume $F: V_n^{(p)} \to V_m^{(p)}$89; Condition A requires even $F: V_n^{(p)} \to V_m^{(p)}$90 and $F: V_n^{(p)} \to V_m^{(p)}$91; the hybrid-sum and code constructions use divisibility conditions such as $F: V_n^{(p)} \to V_m^{(p)}$92, homogeneity constraints $F: V_n^{(p)} \to V_m^{(p)}$93, and dual-scaling congruences $F: V_n^{(p)} \to V_m^{(p)}$94 (Wang et al., 2023, Heng et al., 29 Jun 2025). The Boolean $F: V_n^{(p)} \to V_m^{(p)}$95-based constructions further require permutations whose sums and inverse sums remain permutations, while some non-$F: V_n^{(p)} \to V_m^{(p)}$96 results require $F: V_n^{(p)} \to V_m^{(p)}$97, property $F: V_n^{(p)} \to V_m^{(p)}$98, and absence of linear structures (Polujan et al., 2023).

Open problems in the literature are correspondingly specific. (Bapić, 2022) asks for a monomial exponent $F: V_n^{(p)} \to V_m^{(p)}$99 such that $u\cdot F$00 yields a vectorial $u\cdot F$01-ary weakly regular bent function whose component duals satisfy $u\cdot F$02. (Polujan et al., 2023) asks for sufficient conditions ensuring that the lifted permutations in the recursive $u\cdot F$03 construction preserve the property of being outside $u\cdot F$04, whether recursive lifting preserves the outside-$u\cdot F$05 property, and how to broaden the admissible $u\cdot F$06 beyond monomial trace choices. (Heng et al., 29 Jun 2025) asks for a classification of vectorial dual-bent functions satisfying Conditions I or II, for extensions of explicit hybrid-sum evaluations to broader mixed-character settings, and for constructions of MWBE codebooks from vectorial dual-bent functions. A plausible implication is that the subject is now less constrained by the existence of isolated examples than by the difficulty of reconciling dual closure, homogeneity, weak regularity, and fiberwise combinatorial regularity within a single explicit family.