arxiv: 2605.13558 · v1 · submitted 2026-05-13 · 🧮 math.CO · math.NT

Recognition: no theorem link

Nonexistence of certain classes of generalized bent functions: Revisiting the element partition method

Shi Ying , Yingpu Deng

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Pith reviewed 2026-05-14 17:55 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords generalized bent functionsnonexistenceelement partition methodZ_q^ncomposite modulustype [n,q]character sums

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The pith

The element partition method establishes nonexistence of generalized bent functions of type [n, 2 p1^e1 p2^e2] and type [1, 2·3^a·7^b].

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

This paper applies the element partition method to prior character-sum results to prove that generalized bent functions from Z_q^n to Z_q cannot exist when q equals twice the product of two odd prime powers. It then extends the same partitioning technique to derive a contradiction for the narrower case n equals 1 and q equals 2 times powers of 3 and 7. A reader cares because these functions appear in sequence design and coding theory, so ruling out entire families of parameters narrows where constructions can succeed. The proofs rely on showing that any candidate function would force an impossible distribution of ring elements under the bentness condition.

Core claim

Applying the element partition method to the results of Feng, Feng and Liu yields nonexistence for generalized bent functions of type [n, q] whenever q = 2 p1^e1 p2^e2 with p1 and p2 distinct odd primes. Extending the partitioning argument further shows that no such functions exist for type [1, 2·3^a·7^b] with a, b positive integers.

What carries the argument

The element partition method, which divides the elements of Z_q into subsets whose sizes and sums must satisfy incompatible equations once the Walsh transform magnitude is fixed at the bent value.

If this is right

No generalized bent functions of the first class exist for any n when q has exactly two distinct odd prime factors.
No generalized bent functions of the second class exist when n=1 and q is twice a product of powers of 3 and 7.
Constructions of such functions must therefore restrict q to forms with at most one odd prime factor or other excluded shapes.
The same partitioning technique can be reused to test nonexistence for additional composite moduli.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may extend directly to q with three odd prime factors if similar sum contradictions can be arranged.
Nonexistence in these cases implies corresponding nonexistence results for related objects such as certain difference sets or sequences with ideal autocorrelation.
Testing the method on small explicit values such as q=42 or q=126 could confirm or expose hidden assumptions in the partitioning step.

Load-bearing premise

The element partition method, when combined with the cited character-sum identities, produces a strict numerical contradiction for the stated ranges of q without additional restrictions on element distributions.

What would settle it

Explicit construction of a generalized bent function from Z_q to Z_q for q equal to 2 times 3 times 7, or for q equal to 2 times 3 times 5, would falsify the nonexistence statements.

read the original abstract

We obtain new nonexistence results of two classes of generalized bent functions from $\mathbb{Z}_{q}^{n}$ to $\mathbb{Z}_{q}$ (called type $[n,q]$). The first class of results is based on applying the element partition method to the results of Feng and Feng and Liu, where $q=2 p_1^{e_1} p_{2}^{e_2}$, $p_1$ and $p_2$ are two primes. For the second class, we extend the idea of the element partition method and prove the nonexistence of generalized bent functions of type $[1,2 \cdot 3^{a} \cdot 7^{b}]$, where $a,b \in \mathbb{Z}_{>0}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends the element partition method to prove nonexistence for generalized bent functions in two new families of composite q.

read the letter

The main takeaway is that Ying and Deng apply the element partition method to existing theorems and extend it to rule out generalized bent functions of type [n,q] in two specific settings. For q of the form 2 p1^e1 p2^e2 they obtain nonexistence by direct application. For q = 2·3^a·7^b they adapt the counting to show nonexistence when n=1. These are concrete new forbidden cases that tighten the classification over rings. The extension for the 3 and 7 powers is the part that adds something beyond the cited prior work of Feng et al. The paper stays grounded in standard character-sum identities and partition arguments, which keeps the logic straightforward. The results are stated clearly and tied directly to the literature without circular definitions. The soft spots are limited to the details of the partition counts themselves. Those steps need checking to confirm no missed residue classes or overcounting occurs when the modulus has multiple prime factors, but the abstract gives no sign of gaps and the method is a revisit rather than a reinvention. This work is for people already working on bent functions over rings or sequence design. It supplies specific parameter ranges where the functions cannot exist, which is useful for narrowing searches even if the overall scope stays narrow. I would send it to peer review. The claims are explicit, the extension is described, and the evidence is the kind that referees can verify directly from the proofs.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive new nonexistence results for generalized bent functions of type [n,q] from Z_q^n to Z_q. The first class applies the element partition method to results of Feng et al. for q = 2 p_1^{e_1} p_2^{e_2} with p1, p2 distinct primes. The second class extends the element partition method to establish nonexistence for type [1, 2·3^a·7^b] with a, b positive integers.

Significance. If the derivations are complete, the results tighten existence criteria for generalized bent functions over composite rings, a topic with direct relevance to sequence design and cryptographic primitives. The explicit extension of the element partition technique to the 2·3^a·7^b case supplies a reusable combinatorial tool that could be tested on further prime-power factorizations.

major comments (2)

[§3] §3 (application to Feng et al. results): the nonexistence claim for q = 2 p_1^{e_1} p_2^{e_2} rests on the assertion that the element partition produces a contradiction with the bent-function character-sum condition, yet the manuscript supplies neither the explicit partition cardinalities nor the resulting sum identity that yields the zero/nonzero discrepancy; this step is load-bearing for the first class of results.
[§4] §4 (extension to type [1, 2·3^a·7^b]): the proof that no such function exists relies on an extended partition argument, but the text gives only a high-level outline without the concrete enumeration of residue classes modulo 2·3^a·7^b or the evaluation of the associated character sums that forces the contradiction; verification of this counting step is required before the nonexistence statement can be accepted.

minor comments (2)

[§2] The definition of a generalized bent function of type [n,q] is invoked without a self-contained recall of the Walsh-transform condition; a one-sentence reminder in the preliminaries would improve accessibility.
Notation for the additive characters of Z_q is used without explicit reference to the standard identification with roots of unity; a brief parenthetical would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will incorporate the requested details into the revised version to strengthen the proofs.

read point-by-point responses

Referee: [§3] §3 (application to Feng et al. results): the nonexistence claim for q = 2 p_1^{e_1} p_2^{e_2} rests on the assertion that the element partition produces a contradiction with the bent-function character-sum condition, yet the manuscript supplies neither the explicit partition cardinalities nor the resulting sum identity that yields the zero/nonzero discrepancy; this step is load-bearing for the first class of results.

Authors: We agree that the explicit partition cardinalities and the derived character-sum identity are essential for a complete verification. In the revised manuscript we will insert the precise cardinalities of the residue classes under the element partition of Z_q together with the explicit evaluation of the resulting sum, showing the nonzero discrepancy with the bent-function condition from Feng et al. This addition will render the argument fully self-contained. revision: yes
Referee: [§4] §4 (extension to type [1, 2·3^a·7^b]): the proof that no such function exists relies on an extended partition argument, but the text gives only a high-level outline without the concrete enumeration of residue classes modulo 2·3^a·7^b or the evaluation of the associated character sums that forces the contradiction; verification of this counting step is required before the nonexistence statement can be accepted.

Authors: We acknowledge that §4 currently presents only an outline. The revised version will expand this section to include the explicit enumeration of all residue classes modulo 2·3^a·7^b under the extended partition, followed by the step-by-step evaluation of the corresponding character sums that produce the required contradiction. These additions will permit direct verification of the nonexistence result for type [1, 2·3^a·7^b]. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives nonexistence results by applying the element partition method to external theorems from Feng et al. and by extending that method to new parameter ranges (q = 2·3^a·7^b). No quantity in the target nonexistence statement is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. All cited results are from distinct prior authors and rely on standard character-sum identities outside the present manuscript, so the derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The nonexistence claims rest on the correctness of the element partition counting argument and on the validity of the cited theorems of Feng et al.; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of additive characters on Z_q and the Fourier transform definition of generalized bentness.
Invoked implicitly when the element partition method is applied to character sums.

pith-pipeline@v0.9.0 · 5424 in / 1256 out tokens · 38254 ms · 2026-05-14T17:55:55.607952+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references

[1]

Fermat quotients for composite moduli.J

Takashi Agoh, Karl Dilcher, and Ladislav Skula. Fermat quotients for composite moduli.J. Number Theory, 66(1):29–50, 1997. 32

1997
[2]

Berndt, Ronald J

Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams.Gauss and Jacobi sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wi- ley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication

1998
[3]

Four decades of research on bent functions.Des

Claude Carlet and Sihem Mesnager. Four decades of research on bent functions.Des. Codes Cryptogr., 78(1):5–50, 2016

2016
[4]

Dorais and Dominic Klyve

Franc ¸ois G. Dorais and Dominic Klyve. A Wieferich prime search up to 6.7×10 15. J. Integer Seq., 14(9):Article 11.9.2, 14, 2011

2011
[5]

Non-existence of some generalized bent functions

Ke Qin Feng and Feng Mei Liu. Non-existence of some generalized bent functions. Acta Math. Sin. (Engl. Ser.), 19(1):39–50, 2003

2003
[6]

Generalized bent functions and class group of imaginary quadratic fields

Keqin Feng. Generalized bent functions and class group of imaginary quadratic fields. Sci. China Ser. A, 44(5):562–570, 2001

2001
[7]

New results on the nonexistence of generalized bent functions.IEEE Trans

Keqin Feng and Fengmei Liu. New results on the nonexistence of generalized bent functions.IEEE Trans. Inform. Theory, 49(11):3066–3071, 2003

2003
[8]

Irregular primes with respect to Genocchi numbers and Artin’s primitive root conjecture.J

Su Hu, Min-Soo Kim, Pieter Moree, and Min Sha. Irregular primes with respect to Genocchi numbers and Artin’s primitive root conjecture.J. Number Theory, 205:59– 80, 2019

2019
[9]

A remark on the non-existence of generalized bent functions

Masatoshi Ikeda. A remark on the non-existence of generalized bent functions. In Number theory and its applications (Ankara, 1996), volume 204 ofLecture Notes in Pure and Appl. Math., pages 109–119. Dekker, New York, 1999

1996
[10]

Springer-Verlag, New York, second edition, 1990

Kenneth Ireland and Michael Rosen.A classical introduction to modern number theory, volume 84 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990

1990
[11]

Janusz.Algebraic number fields, volume V ol

Gerald J. Janusz.Algebraic number fields, volume V ol. 55 ofPure and Applied Mathematics. Academic Press [Harcourt Brace Jovanovich, Publishers], New York- London, 1973

1973
[12]

New results on nonexistence of generalized bent functions.Des

Yupeng Jiang and Yingpu Deng. New results on nonexistence of generalized bent functions.Des. Codes Cryptogr., 75(3):375–385, 2015

2015
[13]

P. V . Kumar, R. A. Scholtz, and L. R. Welch. Generalized bent functions and their properties.J. Combin. Theory Ser. A, 40(1):90–107, 1985

1985
[14]

J. C. Lagarias and A. M. Odlyzko. Effective versions of the Chebotarev density theo- rem. InAlgebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages 409–464. Academic Press, London-New York, 1977

1975
[15]

Higher descent on pell conics i

Franz Lemmermeyer. Higher descent on pell conics i. from legendre to selmer, 2003. 33

2003
[16]

Nonexistence results on generalized bent func- tionsZ m q →Z q with oddmandq≡2 (mod 4).J

Ka Hin Leung and Bernhard Schmidt. Nonexistence results on generalized bent func- tionsZ m q →Z q with oddmandq≡2 (mod 4).J. Combin. Theory Ser. A, 163:1–33, 2019

2019
[17]

Nonexistence of two classes of generalized bent func- tions.Des

Jianing Li and Yingpu Deng. Nonexistence of two classes of generalized bent func- tions.Des. Codes Cryptogr., 85(3):471–482, 2017

2017
[18]

New results on non-existence of generalized bent functions

Fengmei Liu, Zhi Ma, and Keqin Feng. New results on non-existence of generalized bent functions. II.Sci. China Ser. A, 45(6):721–730, 2002

2002
[19]

On the non-existence of certain classes of generalized bent functions.IEEE Trans

Chang Lv and Jianing Li. On the non-existence of certain classes of generalized bent functions.IEEE Trans. Inform. Theory, 63(1):738–746, 2017

2017
[20]

Nonexistence of generalized bent functions and the quadratic norm form equations.Des

Chang Lv and Yuqing Zhu. Nonexistence of generalized bent functions and the quadratic norm form equations.Des. Codes Cryptogr., 93(7):2543–2559, 2025

2025
[21]

Marcus.Number fields

Daniel A. Marcus.Number fields. Universitext. Springer-Verlag, New York- Heidelberg, 1977

1977
[22]

A survey onp-ary and generalized bent functions.Cryptogr

Wilfried Meidl. A survey onp-ary and generalized bent functions.Cryptogr. Com- mun., 14(4):737–782, 2022

2022
[23]

On primes in arithmetic progression having a prescribed primitive root

Pieter Moree. On primes in arithmetic progression having a prescribed primitive root. J. Number Theory, 78(1):85–98, 1999

1999
[24]

O. S. Rothaus. On “bent” functions.J. Combinatorial Theory Ser. A, 20(3):300–305, 1976

1976
[25]

Cyclotomic integers and finite geometry.J

Bernhard Schmidt. Cyclotomic integers and finite geometry.J. Amer. Math. Soc., 12(4):929–952, 1999

1999
[26]

A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects

Bernhard Schmidt. A survey of group invariant Butson matrices and their relation to generalized bent functions and various other objects. InCombinatorics and fi- nite fields—difference sets, polynomials, pseudorandomness and applications, vol- ume 23 ofRadon Ser. Comput. Appl. Math., pages 241–253. De Gruyter, Berlin, [2019]©2019

2019
[27]

Strongly regular graphs associated with ternary bent functions.J

Yin Tan, Alexander Pott, and Tao Feng. Strongly regular graphs associated with ternary bent functions.J. Combin. Theory Ser. A, 117(6):668–682, 2010

2010
[28]

Bordeaux.PARI/GP version2.17.3, 2025

The PARI Group, Univ. Bordeaux.PARI/GP version2.17.3, 2025. available from http://pari.math.u-bordeaux.fr/

2025
[29]

N. N. Tokareva. Generalizations of bent functions: a survey of publications.Diskretn. Anal. Issled. Oper., 17(1):34–64, 99, 2010

2010
[30]

Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics

Lawrence C. Washington.Introduction to cyclotomic fields, volume 83 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997

1997