---
title: Deep Particular Partitions in Finite Vector Spaces
url: https://www.emergentmind.com/topics/deep-particular-partitions
type: topic
---

# Deep Particular Partitions in Finite Vector Spaces

A deep particular partition, commonly termed a "bent partition," is a structured partition of the finite vector space $V_n^{(p)}$ (where $n$ is an even positive integer and $p$ is a prime) that underpins the construction of bent functions, vectorial bent functions, partial difference sets, and association schemes. Specifically, a partition $\Gamma = \{A_1,\ldots,A_K\}$ of $V_n^{(p)}$ is called bent if every $p$-ary function $f:V_n^{(p)} \to \mathbb{F}_p$ whose preimages match exactly $K/p$ blocks of $\Gamma$ is a bent function, characterized by Walsh transform values of constant magnitude $p^{n/2}$ for all $a \in V_n^{(p)}$ [2509.16911]. Bent partitions systematically generalize classical bent and vectorial bent functions by allowing rigorous combinatorial and algebraic characterizations, including Hadamard matrix criteria in the binary case.

## 1. Definition and Fundamental Properties

Let $V_n^{(p)}$ denote an $n$-dimensional vector space over $\mathbb{F}_p$ with $n$ even. A bent partition $\Gamma$ of $V_n^{(p)}$ consists of disjoint subsets $\{A_1,\dots,A_K\}$ covering the space; the depth $K=|\Gamma|$ refers to the cardinality of the partition. The requisite property is that any $p$-coloring of $\Gamma$ assigning each color exactly $K/p$ times yields a bent $p$-ary function $f$; thus, partition properties and bentness of all induced functions are tightly coupled.

Equivalently, for an index map $F:V_n^{(p)} \to V_m^{(p)}$ (with $m = \log_p K$ and $F(x)=i$ iff $x\in A_i$), $\Gamma$ is bent if and only if $F$ is vectorial bent and all balanced projections (i.e., any permutation or balanced linear combination of its components) remain bent [2509.16911, Theorem 1]. 

The seminal open problem, originally formulated by Anbar–Meidl (2022), asks whether every bent partition's depth $K$ must always be a power of $p$. To date, all known constructions satisfy $K=p^m$ for some integer $m$.

## 2. Regularity, Weak Regularity, and the Depth Problem

A $p$-ary bent function $f$ is called weakly regular if its Walsh transform has the form $W_f(a) = \epsilon\, p^{n/2}\, \zeta_p^{f^*(a)}$ for each $a$, with $\epsilon \in \{\pm 1\}$ and $f^*$ the dual. If $\epsilon=+1$, $f$ is regular; otherwise, it is only weakly regular. A bent partition is said to be in class "WBP" (*Editor's term*: Weakly/Regular Bent Partition) if all bent functions generated by it are either all regular or all weakly regular, but not a mixture [2509.16911].

The depth–power–of–$p$ theorem asserts that for any WBP-class partition $\Gamma$ of $V_n^{(p)}$, the depth $K$ must be a power of $p$ [2509.16911, Theorem 3]. This result conclusively determines the depth for a large class of bent partitions, and in particular ensures that every Boolean ($p=2$) bent partition has depth a power of $2$.

## 3. Structure Theorems and Proof Outline

The proof of the depth–power–of–$p$ result for WBP-class partitions relies on algebraic characterizations of weakly regular bent functions and the analysis of their duals. The dual $f^*$ of a $p$-ary bent function is constrained by the "c-form" functional identity:
$$
c \cdot f^*(c^{-1}x) = \frac{(c+1)f^*(x) + (c-1)f^*(-x)}{2}, \quad \forall\,c\in\mathbb{F}_p^*,\,x\in V_n^{(p)}.
$$
This identity yields strong structural constraints on the algebraic form of $f^*$ and character sums $\chi_a(D_{f,i})$, where $D_{f,i}$ are the blocks of the partition determined by $f$ [2509.16911, Theorem 2].

Specifically, for each $a$,
$$
\chi_a(D_{f,i}) = p^{n-1}\delta_0(a) + \epsilon\, p^{n/2-1}\, \zeta_p^{h(-a)}\, ( p\,\delta_{g(-a)}(i) - 1 )
$$
where $h$ and $g$ are certain maps determined by $f$ and its dual. Analyzing the range and integrality of these sums constrains $K$ to divide $p^{n/2}$, which, when true for all block unions, mandates $K$ be a power of $p$.

## 4. Constructions: Dual-Bent, Non-Dual-Bent, and Secondary Operations

Two principal construction paradigms for bent partitions are established: those arising from vectorial dual-bent functions and those not directly corresponding to dual-bent maps.

### Vectorial Dual-Bent-Based Partitions

Suppose $F:V_n^{(p)}\to V_m^{(p)}$ is vectorial bent, with all scalar components $F_c$ weakly regular and of the same duality sign $\epsilon$. If one can find maps $G:V_n^{(p)}\to V_m^{(p)}$ and $h:V_n^{(p)}\to\mathbb{F}_p$ such that for $c\neq0$
$$
(F_c)^*(x) = G_c(x) + h(x), \quad\forall x,
$$
then the preimage partition $\Gamma = \{F^{-1}(i)\}$ is bent of class WBP [2509.16911, Theorem 4].

This property is fully expressible in terms of generalized Hadamard matrices $H_c=[\zeta_p^{F_c(x-y)}]_{x,y}$: all such matrices must be of weakly-regular type and the mixed products $H_d H_e^* H_{d-e}^*$ must be equal for all $d,e$.

### Non-Dual-Bent Constructions and Twisted Maiorana-McFarland Forms

For certain vectorial bent $F$, if all $F_c$ are regular (or all weakly regular) and, for $d\neq-e$, the functions $(F_d)^*+(F_e)^*-(F_{d+e})^*$ are all equal to a nonzero $h(x)$, then the blocks $\Gamma = F^{-1}$(i) form a bent partition not arising from any dual-bent $F^*$ [2509.16911, Corollary 1]. 

Infinite families can be constructed using "twisted Maiorana-McFarland" forms:
$$
F(x,y) = Tr_m^n(x\, \pi(y)) + G(y)
$$
with compatible homogeneity and further conditions, yielding partitions beyond the dual-bent scope [2509.16911, Proposition 3]. Additional secondary constructions combine smaller building blocks $R^{(i)}$ or pairs $R, R'$ to form larger vectorial bent functions $F(x,y)$ fulfilling the sufficient condition above.

### Table: Construction Types and Core Properties

| Construction Type                    | Conditions                                 | Key Property                   |
|--------------------------------------|---------------------------------------------|-------------------------------|
| Vectorial dual-bent (⋆ holds, $h=0$) | $F$ dual-bent, balanced $K$                | WBP, $K=p^m$, Hadamard structure |
| Non-dual-bent, "twisted"             | $h\neq 0$ in (⋆), twisted forms            | WBP (not dual-bent), $K=p^m$    |
| Secondary constructions              | Gluing $R^{(i)}$, $R, R'$; balanced bilinear $K$ | Combinatorial families, flexible depth |

## 5. New Families of Vectorial Dual-Bent Functions

Whenever $h(x)=0$ in the duality condition, $F$ is a vectorial dual-bent function. New families can be synthesized by combining two dual-bent functions $R$ and $R'$ using a balanced bilinear form $K$ on their ranges:
$$
F(x,y) = K(R(x), R'(y))
$$
By carefully ensuring that the $h, h'$ constants associated to $R$ and $R'$ have opposite sign, their contributions to the dual vanish, guaranteeing the dual-bent structure of $F$ [2509.16911, Theorem 6]. This operation recovers known Boolean vectorial bent forms and yields new exemplars through targeted substitutions for the building blocks.

## 6. Binary Case and Hadamard Matrix Characterization

For $V_n^{(2)}$, the bent partition criterion becomes purely combinatorial. Let $F:V_n^{(2)} \to V_m^{(2)}$. Then $\Gamma = \{F^{-1}(i)\}$ is bent if and only if the $K-1$ matrices
$$
H_c = [(-1)^{F_c(x\oplus y)}]_{x,y}
$$
are real Hadamard matrices and the triple products $H_d H_e H_{d+e}$ are identical for all $d\ne e$ [2509.16911, Theorem 5]. This implies $K=2^m$ must be a power of $2$, aligning with the existence constraint for Hadamard matrices.

## 7. Open Problems and Future Directions

The depth–power–of–$p$ theorem settles the possible values of $K$ for WBP-class partitions, but the existence and classification of bent partitions outside WBP—those generating a mix of (non-)regular or non-weakly-regular bent functions—remains open. The sufficiency (and possible necessity) of the duality condition $(\star)$ and the Hadamard-matrix criteria provide a practical and conceptual framework for constructing further examples [2509.16911, Remark 2].

Secondary construction techniques, such as gluing smaller vectorial bent functions or using complete permutations, suggest broad avenues for systematically generating new families and a potential classification up to isomorphism. The identification and analysis of non-WBP bent partitions represent an active line of research.

Source: https://www.emergentmind.com/topics/deep-particular-partitions