Deep Particular Partitions in Finite Vector Spaces

• Deep particular partitions are structured partitions of finite vector spaces that produce bent functions with constant Walsh transform magnitude.
• They enable rigorous analysis of bent and vectorial bent functions by linking partition properties with Hadamard matrix criteria and balanced projections.
• Key constructions include vectorial dual-bent and twisted Maiorana-McFarland forms, motivating ongoing research into non-WBP bent partitions.

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 $n$0 (Wang et al., 21 Sep 2025). 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 $n$1 denote an $n$2-dimensional vector space over $n$3 with $n$4 even. A bent partition $n$5 of $n$6 consists of disjoint subsets $n$7 covering the space; the depth $n$8 refers to the cardinality of the partition. The requisite property is that any $n$9-coloring of $p$0 assigning each color exactly $p$1 times yields a bent $p$2-ary function $p$3; thus, partition properties and bentness of all induced functions are tightly coupled.

Equivalently, for an index map $p$4 (with $p$5 and $p$6 iff $p$7), $p$8 is bent if and only if $p$9 is vectorial bent and all balanced projections (i.e., any permutation or balanced linear combination of its components) remain bent [(Wang et al., 21 Sep 2025), Theorem 1].

The seminal open problem, originally formulated by Anbar–Meidl (2022), asks whether every bent partition's depth $\Gamma = \{A_1,\ldots,A_K\}$0 must always be a power of $\Gamma = \{A_1,\ldots,A_K\}$1. To date, all known constructions satisfy $\Gamma = \{A_1,\ldots,A_K\}$2 for some integer $\Gamma = \{A_1,\ldots,A_K\}$3.

2. Regularity, Weak Regularity, and the Depth Problem

A $\Gamma = \{A_1,\ldots,A_K\}$4-ary bent function $\Gamma = \{A_1,\ldots,A_K\}$5 is called weakly regular if its Walsh transform has the form $\Gamma = \{A_1,\ldots,A_K\}$6 for each $\Gamma = \{A_1,\ldots,A_K\}$7, with $\Gamma = \{A_1,\ldots,A_K\}$8 and $\Gamma = \{A_1,\ldots,A_K\}$9 the dual. If $V_n^{(p)}$0, $V_n^{(p)}$1 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 (Wang et al., 21 Sep 2025).

The depth–power–of–$V_n^{(p)}$2 theorem asserts that for any WBP-class partition $V_n^{(p)}$3 of $V_n^{(p)}$4, the depth $V_n^{(p)}$5 must be a power of $V_n^{(p)}$6 [(Wang et al., 21 Sep 2025), Theorem 3]. This result conclusively determines the depth for a large class of bent partitions, and in particular ensures that every Boolean ($V_n^{(p)}$7) bent partition has depth a power of $V_n^{(p)}$8.

3. Structure Theorems and Proof Outline

The proof of the depth–power–of–$V_n^{(p)}$9 result for WBP-class partitions relies on algebraic characterizations of weakly regular bent functions and the analysis of their duals. The dual $p$0 of a $p$1-ary bent function is constrained by the "c-form" functional identity:

$p$2

This identity yields strong structural constraints on the algebraic form of $p$3 and character sums $p$4, where $p$5 are the blocks of the partition determined by $p$6 [(Wang et al., 21 Sep 2025), Theorem 2].

Specifically, for each $p$7,

$p$8

where $p$9 and $f:V_n^{(p)} \to \mathbb{F}_p$0 are certain maps determined by $f:V_n^{(p)} \to \mathbb{F}_p$1 and its dual. Analyzing the range and integrality of these sums constrains $f:V_n^{(p)} \to \mathbb{F}_p$2 to divide $f:V_n^{(p)} \to \mathbb{F}_p$3, which, when true for all block unions, mandates $f:V_n^{(p)} \to \mathbb{F}_p$4 be a power of $f:V_n^{(p)} \to \mathbb{F}_p$5.

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 \mathbb{F}_p$6 is vectorial bent, with all scalar components $f:V_n^{(p)} \to \mathbb{F}_p$7 weakly regular and of the same duality sign $f:V_n^{(p)} \to \mathbb{F}_p$8. If one can find maps $f:V_n^{(p)} \to \mathbb{F}_p$9 and $K/p$0 such that for $K/p$1

$K/p$2

then the preimage partition $K/p$3 is bent of class WBP [(Wang et al., 21 Sep 2025), Theorem 4].

This property is fully expressible in terms of generalized Hadamard matrices $K/p$4: all such matrices must be of weakly-regular type and the mixed products $K/p$5 must be equal for all $K/p$6.

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

For certain vectorial bent $K/p$7, if all $K/p$8 are regular (or all weakly regular) and, for $K/p$9, the functions $\Gamma$0 are all equal to a nonzero $\Gamma$1, then the blocks $\Gamma$2(i) form a bent partition not arising from any dual-bent $\Gamma$3 [(Wang et al., 21 Sep 2025), Corollary 1].

Infinite families can be constructed using "twisted Maiorana-McFarland" forms:

$\Gamma$4

with compatible homogeneity and further conditions, yielding partitions beyond the dual-bent scope [(Wang et al., 21 Sep 2025), Proposition 3]. Additional secondary constructions combine smaller building blocks $\Gamma$5 or pairs $\Gamma$6 to form larger vectorial bent functions $\Gamma$7 fulfilling the sufficient condition above.

Table: Construction Types and Core Properties

Construction Type	Conditions	Key Property
Vectorial dual-bent (⋆ holds, $\Gamma$8)	$\Gamma$9 dual-bent, balanced $p^{n/2}$0	WBP, $p^{n/2}$1, Hadamard structure
Non-dual-bent, "twisted"	$p^{n/2}$2 in (⋆), twisted forms	WBP (not dual-bent), $p^{n/2}$3
Secondary constructions	Gluing $p^{n/2}$4, $p^{n/2}$5; balanced bilinear $p^{n/2}$6	Combinatorial families, flexible depth

5. New Families of Vectorial Dual-Bent Functions

Whenever $p^{n/2}$7 in the duality condition, $p^{n/2}$8 is a vectorial dual-bent function. New families can be synthesized by combining two dual-bent functions $p^{n/2}$9 and $n$00 using a balanced bilinear form $n$01 on their ranges:

$n$02

By carefully ensuring that the $n$03 constants associated to $n$04 and $n$05 have opposite sign, their contributions to the dual vanish, guaranteeing the dual-bent structure of $n$06 [(Wang et al., 21 Sep 2025), 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 $n$07, the bent partition criterion becomes purely combinatorial. Let $n$08. Then $n$09 is bent if and only if the $n$10 matrices

$n$11

are real Hadamard matrices and the triple products $n$12 are identical for all $n$13 [(Wang et al., 21 Sep 2025), Theorem 5]. This implies $n$14 must be a power of $n$15, aligning with the existence constraint for Hadamard matrices.

7. Open Problems and Future Directions

The depth–power–of–$n$16 theorem settles the possible values of $n$17 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 $n$18 and the Hadamard-matrix criteria provide a practical and conceptual framework for constructing further examples [(Wang et al., 21 Sep 2025), 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.