Bent Partitions in Finite Vector Spaces

• Bent partitions are partitions of finite vector spaces over prime fields that guarantee every balanced labeling yields a bent function with a precise Walsh spectrum.
• They are characterized by a depth invariant and exhibit strong links to vectorial dual-bent functions, Hadamard matrices, and affine spread constructions.
• The theory extends to s-plateaued and strong bent partitions, revealing intricate connections between Fourier spectral rigidity, combinatorial designs, and hypergraph frameworks.

Searching arXiv for papers on bent partitions and closely related constructions. arXiv search query: "bent partitions" Bent partitions are partitions of finite vector spaces over prime fields that force bentness under every balanced block labeling. In the current literature, the dominant definition takes a partition $\Gamma=\{A_i,1\le i\le K\}$ of $V_n^{(p)}$, with $n$ even and $p\mid K$, and calls $\Gamma$ a bent partition of depth $K$ when every $p$-ary function $f:V_n^{(p)}\to\mathbb F_p$ for which each $j\in\mathbb F_p$ has exactly $K/p$ of the sets $V_n^{(p)}$0 in its preimage set is bent; equivalently, bent partitions are exactly the $V_n^{(p)}$1-plateaued partitions in the more general theory of $V_n^{(p)}$2-plateaued partitions (Wang et al., 26 Jun 2026). Closely related but non-equivalent usages include strong bent partitions of $V_n^{(p)}$3, which are formulated through a distinguished $V_n^{(p)}$4-dimensional affine subspace and a strong Walsh pattern, and affine-subspace partition frameworks in which partitions into $V_n^{(p)}$5-dimensional affine subspaces act as a construction resource for bent functions rather than as bent partitions in the strict balanced-labeling sense (Potapov et al., 16 Feb 2026).

1. Definition, ambient framework, and terminological scope

The standard ambient object is the $V_n^{(p)}$6-dimensional vector space $V_n^{(p)}$7 over $V_n^{(p)}$8, equipped with a fixed nondegenerate inner product $V_n^{(p)}$9. For a $n$0-ary function $n$1, the Walsh transform is

$n$2

and $n$3 is bent when $n$4 for all $n$5. In the $n$6-plateaued formalism, bent functions are precisely the $n$7-plateaued functions, so the passage from $n$8-plateaued partitions to bent partitions is simply the specialization $n$9 (Wang et al., 26 Jun 2026).

Three nearby notions now coexist in the literature.

Notion	Ambient object	Defining feature
Bent partition	$p\mid K$0	Every balanced labeling of blocks by $p\mid K$1 yields a bent function
Strong bent partition	$p\mid K$2	A stronger Walsh-spectral condition with a distinguished $p\mid K$3-subspace
Affine $p\mid K$4-subspace partition	$p\mid K$5 or $p\mid K$6	Blocks are affine $p\mid K$7-flats used to construct bent functions

This distinction matters. In the strict modern sense, a bent partition is a universal combinatorial object: all balanced assignments of values to blocks must produce bent functions. By contrast, partitions into affine $p\mid K$8-subspaces typically appear as support or fiber decompositions inside specific bent constructions, and strong bent partitions belong to a more specialized binary theory tied to convolution, Fourier duality, and subspace hypergraphs (Wang et al., 21 Sep 2025, Potapov et al., 16 Feb 2026).

2. Structural constraints, depth, and the spread problem

The main global invariant of a bent partition is its depth $p\mid K$9. A central open problem asks whether the depth of any bent partition of $\Gamma$0 must always be a power of $\Gamma$1. A substantial partial resolution is now known: if all bent functions generated by a bent partition are regular, or all are weakly regular but not regular, then the depth must be a power of $\Gamma$2; in particular, every bent partition of $\Gamma$3 has depth a power of $\Gamma$4 (Wang et al., 21 Sep 2025). This isolates the unresolved cases to partitions generating non-weakly regular bent functions, or mixed regularity types.

The same work gives a sharp criterion in the depth-$\Gamma$5 case. If $\Gamma$6 is odd and $\Gamma$7 is a weakly regular bent function, then its level-set partition is a bent partition precisely when its dual satisfies

$\Gamma$8

which constrains the even and odd parts of $\Gamma$9 under scalar multiplication (Wang et al., 21 Sep 2025). This is one of the most explicit known functional criteria for a partition to be bent.

The spread problem concerns maximal depth. For $K$0, a classical construction starts from a spread $K$1 of $K$2 and forms a bent partition of depth $K$3 by taking one block $K$4 and the remaining blocks $K$5. The open question is whether every bent partition of depth $K$6 must arise in this way. For symmetric partitions, the answer is partially affirmative: under the symmetry hypothesis $K$7, and with the additional ternary exclusion stated in the paper, a bent partition of depth $K$8 must be obtained from spreads (Wang et al., 26 Jun 2026).

The general $K$9-plateaued theory sharpens this picture. It shows that bent partitions sit inside a broader family that is structurally more complicated, and it derives cardinality restrictions for the blocks. It also proves that, for $p$0, the preimage partition of a symmetric $p$1-ary $p$2-plateaued function $p$3 with $p$4 is an $p$5-plateaued partition if and only if $p$6 is of $p$7-form, with $p$8 even; the bent-partition statement is the specialization $p$9 (Wang et al., 26 Jun 2026).

3. Vectorial dual-bent functions, binary characterizations, and Hadamard criteria

A major organizing principle is that bent partitions of $f:V_n^{(p)}\to\mathbb F_p$0-power depth are tightly linked to vectorial bent and vectorial dual-bent functions. If $f:V_n^{(p)}\to\mathbb F_p$1 is a partition of $f:V_n^{(p)}\to\mathbb F_p$2, one may define a vectorial map $f:V_n^{(p)}\to\mathbb F_p$3 by $f:V_n^{(p)}\to\mathbb F_p$4 whenever $f:V_n^{(p)}\to\mathbb F_p$5. For odd $f:V_n^{(p)}\to\mathbb F_p$6, bent partitions satisfying Condition C are equivalent to vectorial dual-bent functions satisfying Condition A; in the binary case, the correspondence is subtler because an arbitrary bent partition only guarantees a decomposition

$f:V_n^{(p)}\to\mathbb F_p$7

with a common offset term $f:V_n^{(p)}\to\mathbb F_p$8, and an additional character-value restriction is needed to force genuine vectorial dual-bentness (Wang et al., 2023).

The binary characterization is exact. A vectorial function $f:V_n^{(p)}\to\mathbb F_p$9, with $j\in\mathbb F_p$0 even and $j\in\mathbb F_p$1, is vectorial dual-bent with identity dual permutation,

$j\in\mathbb F_p$2

if and only if its fiber partition $j\in\mathbb F_p$3 is a bent partition and

$j\in\mathbb F_p$4

for every nonzero $j\in\mathbb F_p$5 and every $j\in\mathbb F_p$6. This gives a precise Fourier signature for the binary bent partitions that arise from vectorial dual-bent functions (Wang et al., 2023).

These partitions admit several equivalent reformulations. In the binary case, the same class of bent partitions is equivalent to a family of regular partial difference sets, to a $j\in\mathbb F_p$7-class amorphic association scheme, to a family of two-weight projective linear codes, and to a family of Hadamard matrices $j\in\mathbb F_p$8 satisfying

$j\in\mathbb F_p$9

For odd $K/p$0, the corresponding equivalences use generalized Hadamard matrices and the subtraction law $K/p$1, where $K/p$2 is the common weak-regularity sign (Wang et al., 2023).

Recent work pushes this further. For general bent partitions of $K/p$3, not only those already known to come from vectorial dual-bent functions, a characterization in terms of Hadamard matrices is now available (Wang et al., 21 Sep 2025). The resulting picture is that binary bent partitions sit at an intersection of Walsh-spectral rigidity, Hadamard-matrix identities, and the combinatorics of translation schemes.

4. Strong bent partitions and the hypergraph-convolution viewpoint

Strong bent partitions form a distinct binary subtheory. Here the ambient object is a partition

$K/p$4

of $K/p$5, where $K/p$6 is an $K/p$7-dimensional affine subspace and the remaining cells have equal size. The defining property is not the universal balanced-labeling condition from $K/p$8-bent partitions, but a stronger Walsh pattern: outside $K/p$9, each Fourier vector has exactly one “large” coordinate and all remaining coordinates equal to a uniform background value. This stronger condition produces a dual strong bent partition

$V_n^{(p)}$00

and the Fourier transforms of the indicators $V_n^{(p)}$01 take the explicit form

$V_n^{(p)}$02

(Potapov et al., 16 Feb 2026).

The principal new interpretation is hypergraph-theoretic. Let $V_n^{(p)}$03 be the $V_n^{(p)}$04-uniform hypergraph with vertex set $V_n^{(p)}$05 and hyperedges $V_n^{(p)}$06 satisfying $V_n^{(p)}$07. Every strong bent partition yields a perfect coloring of $V_n^{(p)}$08 with color classes

$V_n^{(p)}$09

The proof runs through multidimensional convolution matrices, Fourier multiplication, and explicit closure relations for convolutions of cell indicators (Potapov et al., 16 Feb 2026).

This places strong bent partitions alongside spreads, partial difference sets, and plateaued-function designs inside a common convolution-eigenfunction framework. It also clarifies that “bent partition” is not a single invariant notion across the literature: strong bent partitions are spectrally stricter and are naturally encoded as perfect colorings of a subspace hypergraph rather than solely as universal balanced-labeling partitions.

5. Affine $V_n^{(p)}$10-subspace and spread-like partitions in bent-function constructions

A second partition-theoretic current in the literature studies partitions into affine or spread-like subspaces that generate bent functions without themselves being bent partitions in the strict universal sense. In generalized Maiorana–McFarland constructions, the decisive object is often a partition of a Boolean space into affine $V_n^{(p)}$11-flats. For functions

$V_n^{(p)}$12

bentness is equivalent to the fibers $V_n^{(p)}$13 forming a partition of $V_n^{(p)}$14 into $V_n^{(p)}$15-dimensional affine subspaces, with $V_n^{(p)}$16 having odd weight on each fiber. For a fixed such partition, there are exactly

$V_n^{(p)}$17

choices of $V_n^{(p)}$18 yielding bent functions. In the case $V_n^{(p)}$19, explicit proper partitions of $V_n^{(p)}$20 into affine $V_n^{(p)}$21-subspaces produce at least $V_n^{(p)}$22 bent functions outside $V_n^{(p)}$23 (Kudin et al., 19 Aug 2025).

The asymptotic counting perspective reaches the same combinatorial core from a different direction. Construction (K) builds bent functions from ordered partitions of $V_n^{(p)}$24 into affine subspaces of dimension $V_n^{(p)}$25, and the case $V_n^{(p)}$26 is asymptotically optimal among fixed-gap instances. In particular, partitions into $V_n^{(p)}$27-dimensional affine subspaces yield the lower bound

$V_n^{(p)}$28

and the number $V_n^{(p)}$29 of partitions of $V_n^{(p)}$30 into affine $V_n^{(p)}$31-subspaces satisfies

$V_n^{(p)}$32

(Potapov et al., 2021).

Spread-like partitions furnish another generalization. In $V_n^{(p)}$33, certain nonlinear partitions $V_n^{(p)}$34 behave like Desarguesian spreads from the viewpoint of bentness: arbitrary labelings by a group of order $V_n^{(p)}$35 produce $V_n^{(p)}$36-bent functions, and the union of exactly $V_n^{(p)}$37 blocks is always the support of a Boolean bent function. In the extremal case $V_n^{(p)}$38, these partitions reduce to the classical spread construction (Meidl et al., 2020).

6. Generalizations, adjacent notions, and current scope

The most systematic generalization is the theory of $V_n^{(p)}$39-plateaued partitions. Bent partitions are exactly the $V_n^{(p)}$40 case, but $V_n^{(p)}$41-plateaued partitions also generate vectorial $V_n^{(p)}$42-plateaued functions and generalized $V_n^{(p)}$43-plateaued functions, and their structure is explicitly described as “much more complicated than bent partitions” (Wang et al., 26 Jun 2026). This broader theory has become the natural ambient framework for current work on depth, symmetry, linear structures, and spread-type extremal cases.

Several adjacent partition-style viewpoints do not define bent partitions proper, but remain structurally relevant. Bent functions linear on the elements of spreads are organized by spread partitions of $V_n^{(p)}$44, and their bentness is equivalent to a line-oval condition in the associated affine plane (Abdukhalikov, 2016). Generalized bent functions $V_n^{(p)}$45 can be characterized as affine spaces of bent or semi-bent Boolean functions, giving a family-level “block” structure rather than a set partition of the ambient space (Hodžić et al., 2016). Non-weakly regular bent functions also induce natural partitions $V_n^{(p)}$46 of $V_n^{(p)}$47; these are used to construct symmetric association schemes of classes $V_n^{(p)}$48, $V_n^{(p)}$49, and $V_n^{(p)}$50, but they are partitions arising from a bent function rather than bent partitions in the balanced-labeling sense (Wei et al., 2024).

A persistent source of ambiguity is therefore terminological. In current usage, “bent partitions” may denote the strict universal partitions of $V_n^{(p)}$51, the stronger binary objects tied to subspace hypergraphs, or the affine-subspace decompositions that feed particular bent constructions. The unifying theme is spectral rigidity under partition data. The differences lie in what is being partitioned—ambient vectors, Walsh supports, spread elements, or affine $V_n^{(p)}$52-flats—and in whether the partition is itself the primary object or a combinatorial device inside a bent-function construction (Wang et al., 21 Sep 2025, Potapov et al., 16 Feb 2026).