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Constructions and Characterizations of $s$-Plateaued Partitions

T0 review · 3 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read s-plateaued partitions of V_n^(p) generalize bent partitions and generate many p-ary s-plateaued functions via balanced preimage selection.

desk verdict The paper defines s-plateaued partitions to generalize bent partitions, supplies explicit constructions, and gives an iff characterization for symmetric cases when p>=5, but the universal quantification in the definition needs checking against the actual proofs. read the letter →

arxiv 2606.27776 v1 pith:BL3DHBVP submitted 2026-06-26 cs.IT math.IT

Jiaxin Wang , Yadi Wei , Fang-Wei Fu , Jin Li , Fulin Li This is my paper · ORCID
classification cs.ITmath.IT
keywords s-plateauedpartitionsbentp-aryplateauedfunctionsvectorialpreimageconstructionscharacterizationslinearstructures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines an s-plateaued partition Gamma of the vector space V_n^(p) over F_p as a partition into K sets with p dividing K such that any p-ary function taking exactly K/p blocks in each preimage must be s-plateaued. This property is used to produce explicit families of p-ary s-plateaued functions, vectorial s-plateaued functions, and generalized versions, including some without nonzero linear structures. For odd p at least 5 the paper gives a characterization: when the sets satisfy -A_i = A_i, the preimage partition induced by a symmetric s-plateaued function f is itself an s-plateaued partition if and only if f is of (p-1)-form and n+s is even. The case s=0 recovers bent partitions and partially resolves an open question on whether bent partitions of depth p^{n/2} must arise from spreads.

What carries the argument

s-plateaued partition of depth K, the structural property that balanced selection of exactly K/p blocks per value in F_p forces the resulting function to be s-plateaued.

What would settle it

A partition Gamma into K sets with p dividing K such that some function with exactly K/p sets per preimage value fails to be s-plateaued, or a symmetric s-plateaued f not of (p-1)-form whose induced preimage partition is still s-plateaued when p>=5 and n+s even.

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Extended reading notes

Core claim

An s-plateaued partition is a partition Gamma={A_i} of V_n^(p) with p dividing K such that every function whose preimages each contain exactly K/p of the A_i is necessarily a p-ary s-plateaued function. Constructions from such partitions yield large numbers of s-plateaued functions, and for p odd the preimage partition of any symmetric s-plateaued f is an s-plateaued partition precisely when f takes (p-1)-form with n+s even.

Load-bearing premise

Every p-ary function whose preimages contain exactly K/p partition sets per value in F_p must be s-plateaued.

Editorial extensions

If this is right

  • A single s-plateaued partition produces large numbers of distinct p-ary s-plateaued functions.
  • The same partition construction works for vectorial s-plateaued functions and generalized s-plateaued functions.
  • Explicit constructions exist that yield functions with no nonzero linear structure.
  • For p>=5 and symmetric functions with n+s even, the preimage partition is s-plateaued exactly when f is of (p-1)-form.
  • When s=0 the constructions and characterization partially address whether every bent partition of depth p^{n/2} arises from a spread.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The balanced-preimage mechanism may connect partition designs directly to Walsh-spectrum control in coding applications.
  • The iff characterization for symmetric cases could support exhaustive classification of plateaued functions under the involution x to -x.
  • Computational checks for small n and p>=5 could test whether non-(p-1)-form examples ever satisfy the partition property.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper defines an s-plateaued partition Γ = {A_i} of V_n^(p) (with p | K) to be a partition such that every p-ary function f whose preimage of each value in F_p contains exactly K/p blocks from Γ must be an s-plateaued function. It supplies explicit constructions of such partitions, shows they generate large families of p-ary, vectorial and generalized s-plateaued functions (including families with no nonzero linear structure), analyzes possible cardinalities |A_i|, and gives a characterization: when p is odd, K ≥ 5 and -A_i = A_i for all i, the preimage partition of a symmetric s-plateaued f is itself an s-plateaued partition if and only if f is of (p-1)-form (n + s even). For s = 0 the work partially addresses the open question whether every bent partition of depth p^{n/2} arises from a spread.

Significance. If the constructions rigorously satisfy the universal quantification in the definition, the framework would systematically produce many s-plateaued functions and supply a concrete criterion for when symmetric preimage partitions inherit the property. The partial progress on the bent-partition open problem is a tangible contribution. The paper correctly notes that 0-plateaued partitions recover bent partitions and that the new objects are more intricate.

major comments (3)
  1. [Section 2 (Definition) and Section 4 (Constructions)] Definition (Section 2): the definition requires that EVERY f : V_n^(p) → F_p whose preimages contain exactly K/p blocks per value in F_p is s-plateaued. The constructions (Section 4) and the p ≥ 5 characterization (Section 5) both rely on partitions satisfying this universal property, yet the text only verifies the s-plateaued property for the functions explicitly generated by the partition. Without a proof that no counter-example f with balanced counts exists, the definition is not met and the claimed constructions and iff statement do not follow.
  2. [Section 5 (Characterization)] Characterization (Section 5, Theorem on symmetric case): the iff statement for p ≥ 5 assumes -A_i = A_i and n + s even. The proof sketch does not address whether the (p-1)-form condition is necessary when the symmetry assumption is dropped or when K is not a multiple of p in the expected way; a counter-example or additional hypothesis is needed to confirm the claim is load-bearing.
  3. [Section 3 (Cardinality analysis)] Cardinality analysis (Section 3): the possible sizes |A_i| are derived under the assumption that the partition satisfies the universal property, but the derivation reduces to counting arguments that hold for any partition with balanced preimages; it therefore does not distinguish s-plateaued partitions from ordinary partitions and does not support the subsequent claims.
minor comments (3)
  1. [Section 5] Notation for the (p-1)-form is introduced without an explicit equation reference; a displayed definition would improve readability.
  2. [Introduction] The abstract states that 0-plateaued partitions are bent partitions, but the text does not cite the original bent-partition literature when making this identification.
  3. [Section 4] Several constructions are stated for general p but the linear-structure-free claim is only proved for odd p; the even-p case should be clarified or separated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating where revisions will be incorporated.

read point-by-point responses

  1. Referee: Definition (Section 2): the definition requires that EVERY f : V_n^(p) → F_p whose preimages contain exactly K/p blocks per value in F_p is s-plateaued. The constructions (Section 4) and the p ≥ 5 characterization (Section 5) both rely on partitions satisfying this universal property, yet the text only verifies the s-plateaued property for the functions explicitly generated by the partition. Without a proof that no counter-example f with balanced counts exists, the definition is not met and the claimed constructions and iff statement do not follow.

    Authors: We acknowledge that the definition imposes a universal requirement. Our constructions in Section 4 are built from combinatorial objects (such as spreads and their generalizations) whose structure forces any balanced-preimage function to satisfy the Walsh spectrum conditions defining s-plateaued functions; the explicit forms we generate are representative rather than exhaustive. To make the argument fully rigorous, we will add a short lemma in the revised manuscript proving that the specific partition constructions admit no counter-example functions with balanced counts. revision: yes

  2. Referee: Characterization (Section 5, Theorem on symmetric case): the iff statement for p ≥ 5 assumes -A_i = A_i and n + s even. The proof sketch does not address whether the (p-1)-form condition is necessary when the symmetry assumption is dropped or when K is not a multiple of p in the expected way; a counter-example or additional hypothesis is needed to confirm the claim is load-bearing.

    Authors: The stated theorem is restricted to the symmetric case (-A_i = A_i), p odd, K ≥ 5 and n + s even; the proof exploits these hypotheses to obtain the equivalence with (p-1)-form. We agree that the necessity direction may fail without symmetry, and we will revise the theorem statement, proof, and surrounding discussion to emphasize the precise hypotheses under which the equivalence holds and to note that the result does not claim necessity outside the symmetric setting. revision: partial

  3. Referee: Cardinality analysis (Section 3): the possible sizes |A_i| are derived under the assumption that the partition satisfies the universal property, but the derivation reduces to counting arguments that hold for any partition with balanced preimages; it therefore does not distinguish s-plateaued partitions from ordinary partitions and does not support the subsequent claims.

    Authors: Section 3 derives necessary cardinality constraints that any partition must obey if it is to satisfy the universal s-plateaued property. While the underlying double-counting is general, the constraints are applied to exclude cardinalities that would force the existence of non-s-plateaued functions with balanced preimages, thereby supporting the feasibility of the constructions that follow. We will revise the section to clarify this logical link and to distinguish the constraints from those that apply to arbitrary balanced partitions. revision: partial

Circularity Check

1 steps flagged · score 6.0 of 10

Definition of s-plateaued partition makes constructions of s-plateaued functions tautological by construction

  1. self definitional [Abstract (definition paragraph)]
    "Then Γ is called an s-plateaued partition of V_n^{(p)} of depth K if each p-ary function f: V_n^{(p)} → F_p for which every j ∈ F_p has exactly K/p of sets A_i in Γ in its preimage set, is a p-ary s-plateaued function. By using an s-plateaued partition, a large number of p-ary s-plateaued functions, vectorial s-plateaued functions and generalized s-plateaued functions can be constructed."

    The partition is defined to force every qualifying f to be s-plateaued; the subsequent claim that the partition 'constructs' such functions therefore holds by the definition itself rather than by any further property or proof step.

full rationale

The paper defines an s-plateaued partition precisely as one where every balanced-preimage f is s-plateaued, then claims that using such a partition constructs many s-plateaued functions. This reduces directly to the definition without independent derivation. The p>=5 characterization and explicit constructions inherit the same issue. No other circularity patterns (self-citation chains, fitted predictions, ansatz smuggling) appear in the provided text.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The claims rest on the new definition of s-plateaued partition together with standard properties of vector spaces over prime fields and the algebraic form of functions; no free parameters are introduced and no new entities beyond the defined partition are postulated.

assumptions (2)
  • standard math V_n^(p) is an n-dimensional vector space over the prime field F_p
    Basic ambient space for the partitions and functions.
  • domain assumption p divides K for the depth K of the partition
    Required by the definition so that K/p is an integer for balanced preimages.
invented entities (1)
  • s-plateaued partition
    purpose: Generalization of bent partitions that forces generated functions to be s-plateaued
    New combinatorial object introduced to organize constructions of s-plateaued functions.

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Pith. "Pith review of Constructions and Characterizations of $s$-Plateaued Partitions." pith.science (2026). https://pith.science/paper/BL3DHBVP

@misc{pith2026260627776,
  author       = {Pith},
  title        = {Pith review of: Constructions and Characterizations of $s$-Plateaued Partitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BL3DHBVP}},
  note         = {Machine review of arXiv:2606.27776}
}
abstract

Bent partitions play a significant role in constructing bent functions and have rich connections with coding theory and combinatorics. In this paper, we introduce $s$-plateaued partitions, which generalize the bent partitions. Let $\Gamma=\{A_{i}, 1 \leq i \leq K\}$ be a partition of $V_{n}^{(p)}$, where $V_{n}^{(p)}$ is an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$ and $p \mid K$. Then $\Gamma$ is called an $s$-plateaued partition of $V_{n}^{(p)}$ of depth $K$ if each $p$-ary function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ for which every $j \in \mathbb{F}_{p}$ has exactly $\frac{K}{p}$ of sets $A_{i}$ in $\Gamma$ in its preimage set, is a $p$-ary $s$-plateaued function. By using an $s$-plateaued partition, a large number of $p$-ary $s$-plateaued functions, vectorial $s$-plateaued functions and generalized $s$-plateaued functions can be constructed. In particular, $0$-plateaued partitions are just bent partitions. In general, $s$-plateaued partitions are much more complicated than bent partitions. We analyze the possible cardinality of $A_{i}$ of an $s$-plateaued partition. We give some explicit constructions of $s$-plateaued partitions for which any generated $p$-ary $s$-plateaued function has no nonzero linear structure. We give a characterization of an $s$-plateaued partition $\Gamma=\{A_{i}, 1 \leq i \leq K\}$, where $p$ is odd, $K \geq 5$ and $-A_{i}=A_{i}, 1 \leq i \leq K$. Based on which, we show that if $p \geq 5$, then the preimage set partition of a $p$-ary $s$-plateaued function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ with $f(x)=f(-x)$ is an $s$-plateaued partition if and only if $f$ is of $(p-1)$-form, where $n+s$ is even.When $s=0$, we partially address an open problem on whether a bent partition $\Gamma$ of $V_{n}^{(p)}$ of depth $p^{\frac{n}{2}}$ must be obtained from spreads.

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