---
title: Bent Partitions in Finite Vector Spaces
url: https://www.emergentmind.com/topics/bent-partitions
type: topic
---

# Bent Partitions in Finite Vector Spaces

Searching arXiv for recent 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 \(A_i\) in its preimage set is bent; equivalently, bent partitions are exactly the \(0\)-plateaued partitions in the more general theory of \(s\)-plateaued partitions [2606.27776]. Closely related but non-equivalent usages include strong bent partitions of \(\mathbb F_2^n\), which are formulated through a distinguished \(n/2\)-dimensional affine subspace and a strong Walsh pattern, and affine-subspace partition frameworks in which partitions into \(2\)-dimensional affine subspaces act as a construction resource for bent functions rather than as bent partitions in the strict balanced-labeling sense [2602.14569].

## 1. Definition, ambient framework, and terminological scope

The standard ambient object is the \(n\)-dimensional vector space \(V_n^{(p)}\) over \(\mathbb F_p\), equipped with a fixed nondegenerate inner product \(\langle\cdot,\cdot\rangle_n\). For a \(p\)-ary function \(f:V_n^{(p)}\to\mathbb F_p\), the Walsh transform is
\[
W_f(a)=\sum_{x\in V_n^{(p)}} \zeta_p^{\,f(x)-\langle a,x\rangle_n},\qquad a\in V_n^{(p)},
\]
and \(f\) is bent when \(|W_f(a)|=p^{n/2}\) for all \(a\in V_n^{(p)}\). In the \(s\)-plateaued formalism, bent functions are precisely the \(0\)-plateaued functions, so the passage from \(s\)-plateaued partitions to bent partitions is simply the specialization \(s=0\) [2606.27776].

Three nearby notions now coexist in the literature.

| Notion | Ambient object | Defining feature |
|---|---|---|
| Bent partition | \(V_n^{(p)}\) | Every balanced labeling of blocks by \(\mathbb F_p\) yields a bent function |
| Strong bent partition | \(\mathbb F_2^n\) | A stronger Walsh-spectral condition with a distinguished \(n/2\)-subspace |
| Affine \(2\)-subspace partition | \(\mathbb F_2^m\) or \(\mathbb F_2^{m+1}\) | Blocks are affine \(2\)-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 \(2\)-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 [2509.16911; 2602.14569].

## 2. Structural constraints, depth, and the spread problem

The main global invariant of a bent partition is its depth \(K\). A central open problem asks whether the depth of any bent partition of \(V_n^{(p)}\) must always be a power of \(p\). 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 \(p\); in particular, every bent partition of \(V_n^{(2)}\) has depth a power of \(2\) [2509.16911]. 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-\(p\) case. If \(p\) is odd and \(f:V_n^{(p)}\to\mathbb F_p\) is a weakly regular bent function, then its level-set partition is a bent partition precisely when its dual satisfies
\[
cf^*(c^{-1}x)=\frac{(c+1)f^*(x)+(c-1)f^*(-x)}{2}, \qquad c\in\mathbb F_p^*,
\]
which constrains the even and odd parts of \(f^*\) under scalar multiplication [2509.16911]. This is one of the most explicit known functional criteria for a partition to be bent.

The spread problem concerns maximal depth. For \(n=2m\), a classical construction starts from a spread \(\{U_0,U_1,\dots,U_{p^m}\}\) of \(V_n^{(p)}\) and forms a bent partition of depth \(p^m\) by taking one block \(U_0\cup U_1\) and the remaining blocks \(U_j^*=U_j\setminus\{0\}\). The open question is whether every bent partition of depth \(p^{n/2}\) must arise in this way. For symmetric partitions, the answer is partially affirmative: under the symmetry hypothesis \(-A_i=A_i\), and with the additional ternary exclusion stated in the paper, a bent partition of depth \(p^{n/2}\) must be obtained from spreads [2606.27776].

The general \(s\)-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 \(p\ge 5\), the preimage partition of a symmetric \(p\)-ary \(s\)-plateaued function \(f\) with \(f(x)=f(-x)\) is an \(s\)-plateaued partition if and only if \(f\) is of \((p-1)\)-form, with \(n+s\) even; the bent-partition statement is the specialization \(s=0\) [2606.27776].

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

A major organizing principle is that bent partitions of \(p\)-power depth are tightly linked to vectorial bent and vectorial dual-bent functions. If \(\Gamma=\{A_i,\ i\in V_m^{(p)}\}\) is a partition of \(V_n^{(p)}\), one may define a vectorial map \(F:V_n^{(p)}\to V_m^{(p)}\) by \(F(x)=i\) whenever \(x\in A_i\). For odd \(p\), 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
\[
(F_c)^*(x)=G_c(x)+\delta_S(x),
\]
with a common offset term \(\delta_S\), and an additional character-value restriction is needed to force genuine vectorial dual-bentness [2309.13395].

The binary characterization is exact. A vectorial function \(F:V_n^{(2)}\to V_m^{(2)}\), with \(n\ge 4\) even and \(2\le m\le n/2\), is vectorial dual-bent with identity dual permutation,
\[
(F_c)^*=(F^*)_c \qquad \text{for all } c\in V_m^{(2)*},
\]
if and only if its fiber partition \(\{D_{F,i},\,i\in V_m^{(2)}\}\) is a bent partition and
\[
\chi_u(D_{F,i})\in\{-2^{n/2-m},\ -2^{n/2-m}+2^{n/2}\}
\]
for every nonzero \(u\) and every \(i\). This gives a precise Fourier signature for the binary bent partitions that arise from vectorial dual-bent functions [2309.13395].

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 \(2^m\)-class amorphic association scheme, to a family of two-weight projective linear codes, and to a family of Hadamard matrices \(H_c\) satisfying
\[
H_cH_d=2^n H_{c+d}.
\]
For odd \(p\), the corresponding equivalences use generalized Hadamard matrices and the subtraction law \(H_cH_d=\epsilon p^{n/2}H_{c-d}\), where \(\epsilon\) is the common weak-regularity sign [2309.13395].

Recent work pushes this further. For general bent partitions of \(V_n^{(2)}\), not only those already known to come from vectorial dual-bent functions, a characterization in terms of Hadamard matrices is now available [2509.16911]. 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
\[
\{U,P_1,\dots,P_{2^k}\}
\]
of \(\mathbb F_2^n\), where \(U\) is an \(n/2\)-dimensional affine subspace and the remaining cells have equal size. The defining property is not the universal balanced-labeling condition from \(V_n^{(p)}\)-bent partitions, but a stronger Walsh pattern: outside \(U^\perp\), 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
\[
\{U^\perp,R_1,\dots,R_{2^k}\},
\]
and the Fourier transforms of the indicators \(\chi_{P_j}\) take the explicit form
\[
\widehat{\chi}_{P_j}=\chi_{R_j}-\frac{1}{2^k}\mathbf 1+\frac{2^{n/2}}{2^k}\chi_{\bar 0}
\]
[2602.14569].

The principal new interpretation is hypergraph-theoretic. Let \(\mathcal H_n\) be the \(3\)-uniform hypergraph with vertex set \(\mathbb F_2^n\setminus\{0\}\) and hyperedges \(\{x,y,z\}\) satisfying \(x+y+z=0\). Every strong bent partition yields a perfect coloring of \(\mathcal H_n\) with color classes
\[
U\setminus\{0\},\ P_1,\dots,P_{2^k}.
\]
The proof runs through multidimensional convolution matrices, Fourier multiplication, and explicit closure relations for convolutions of cell indicators [2602.14569].

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 \(2\)-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 \(2\)-flats. For functions
\[
f(x,y)=x\cdot \phi(y)+h(y),\qquad x\in\mathbb F_2^{m-1},\ y\in\mathbb F_2^{m+1},
\]
bentness is equivalent to the fibers \(\{\phi^{-1}(a)\mid a\in\mathbb F_2^{m-1}\}\) forming a partition of \(\mathbb F_2^{m+1}\) into \(2\)-dimensional affine subspaces, with \(h\) having odd weight on each fiber. For a fixed such partition, there are exactly
\[
2^{3\cdot 2^{m-1}}
\]
choices of \(h\) yielding bent functions. In the case \(2m=8\), explicit proper partitions of \(\mathbb F_2^5\) into affine \(2\)-subspaces produce at least \(2^{78}\) bent functions outside \(\mathcal M^\#\) [2508.14265].

The asymptotic counting perspective reaches the same combinatorial core from a different direction. Construction (K) builds bent functions from ordered partitions of \(\mathbb F_2^{n_2}\) into affine subspaces of dimension \(n_2-n_1\), and the case \(n_2-n_1=2\) is asymptotically optimal among fixed-gap instances. In particular, partitions into \(2\)-dimensional affine subspaces yield the lower bound
\[
\log_2 b_n \ge \frac{3n}{2}\,2^{n/2} - 2\log_2(e)\,2^{n/2} + O(2^{n/2}),
\]
and the number \(N_m\) of partitions of \(\mathbb F_2^m\) into affine \(2\)-subspaces satisfies
\[
\log_2 N_m = \frac m2\,2^m + \Theta(2^m)
\]
[2108.00232].

Spread-like partitions furnish another generalization. In \(\mathbb F_{2^m}\times\mathbb F_{2^m}\), certain nonlinear partitions \(\mathcal T_1,\mathcal T_2\) behave like Desarguesian spreads from the viewpoint of bentness: arbitrary labelings by a group of order \(2^k\) produce \(\mathbb Z_{2^k}\)-bent functions, and the union of exactly \(2^{k-1}\) blocks is always the support of a Boolean bent function. In the extremal case \(k=m\), these partitions reduce to the classical spread construction [2009.11019].

## 6. Generalizations, adjacent notions, and current scope

The most systematic generalization is the theory of \(s\)-plateaued partitions. Bent partitions are exactly the \(s=0\) case, but \(s\)-plateaued partitions also generate vectorial \(s\)-plateaued functions and generalized \(s\)-plateaued functions, and their structure is explicitly described as “much more complicated than bent partitions” [2606.27776]. 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 \(\mathbb F_2^{2m}\), and their bentness is equivalent to a line-oval condition in the associated affine plane [1609.03138]. Generalized bent functions \(V_n\to\mathbb Z_{2^k}\) 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 [1605.05713]. Non-weakly regular bent functions also induce natural partitions \(P,D,T,U,V\) of \(\mathbb F_p^n\); these are used to construct symmetric association schemes of classes \(2p\), \((2p+1)\), and \(\frac{3p+1}{2}\), but they are partitions arising from a bent function rather than bent partitions in the balanced-labeling sense [2404.05251].

A persistent source of ambiguity is therefore terminological. In current usage, “bent partitions” may denote the strict universal partitions of \(V_n^{(p)}\), 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 \(2\)-flats—and in whether the partition is itself the primary object or a combinatorial device inside a bent-function construction [2509.16911; 2602.14569].

Source: https://www.emergentmind.com/topics/bent-partitions