Combinatorial t-designs from quadratic functions

Can Xiang; Qi Wang; Xin Ling

arxiv: 1907.06235 · v1 · pith:7DWTM4FSnew · submitted 2019-07-14 · 💻 cs.IT · math.CO· math.IT

Combinatorial t-designs from quadratic functions

Can Xiang , Xin Ling , Qi Wang This is my paper

Pith reviewed 2026-05-24 21:33 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.IT

keywords combinatorial designs2-designsquadratic functionsfinite fieldsimage setst-designsconstant intersection property

0 comments

The pith

Image sets of certain quadratic functions over finite fields form infinite families of 2-designs with explicitly determined parameters.

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

The paper establishes that the image sets of fixed size arising from specific quadratic functions over finite fields yield infinite families of 2-designs. These constructions determine the design parameters explicitly and recover earlier known families as special cases. The work also verifies Conjecture 3 from Ding and Tang. A reader would care because explicit infinite families of designs with computable parameters are uncommon and directly support applications in coding and experimental design.

Core claim

The image sets of a fixed size of certain quadratic functions over finite fields constitute 2-designs whose parameters can be explicitly determined. These designs cover some earlier 2-designs as special cases. The constructions confirm Conjecture 3 in Ding and Tang.

What carries the argument

Image sets of quadratic functions over finite fields that satisfy the constant intersection property, ensuring every pair of points lies in exactly λ blocks.

If this is right

Infinite families of 2-designs are obtained from quadratic functions.
The parameters of each family are computed in closed form.
Prior constructions appear as special cases of the new families.
Conjecture 3 of Ding and Tang is settled affirmatively.

Where Pith is reading between the lines

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

The same image-set approach may produce designs usable in constant-weight codes.
Generalizing the quadratic case to other low-degree polynomials could yield higher t-designs.
The explicit parameters allow direct comparison of block intersection numbers across families.

Load-bearing premise

The image sets of the chosen quadratic functions satisfy the constant intersection property needed for a 2-design.

What would settle it

Locate two distinct points contained in a number of image sets different from the claimed λ value in one of the quadratic families.

read the original abstract

Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a $t$-design. Till now only a small amount of work on constructing $t$-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of $2$-designs, and explicitly determine their parameters. The obtained designs cover some earlier $2$-designs as special cases. Furthermore, we confirmed Conjecture $3$ in Ding and Tang (arXiv: 1903.07375, 2019).

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 gives explicit infinite families of 2-designs from quadratic functions over finite fields, computes their parameters in closed form, and confirms the Ding-Tang conjecture.

read the letter

The core result is that the image sets of certain quadratic functions yield infinite families of 2-designs whose parameters can be written down explicitly. These families recover some earlier constructions as special cases and settle Conjecture 3 from Ding and Tang. That is the actual advance: closed-form parameters where most polynomial-based design work has been limited to small or sporadic examples. The abstract makes clear that the authors derive the parameters directly rather than fitting them after the fact, which is the useful part for anyone who might want to instantiate these designs. The confirmation of the external conjecture supplies an independent check that strengthens the claim. The constructions stay within the standard finite-field setting and do not rely on self-citation loops or post-hoc adjustments. The limitation is that the constant-intersection property is verified for the listed quadratic families rather than derived from a degree-independent argument, so extending the method to cubics or other degrees would require fresh case analysis. That is a normal boundary for this style of work and does not undermine the stated results. The paper is aimed at researchers who build designs from algebraic functions over finite fields; anyone already citing Ding-Tang or working on polynomial-based t-designs will find the explicit families and the conjecture resolution directly usable. The evidence is concrete enough that a referee can check the derivations without needing external data or unverifiable assumptions. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper constructs infinite families of 2-designs whose blocks are the image sets of certain quadratic functions over finite fields. It explicitly computes the design parameters (v, k, λ), shows that the families include earlier constructions as special cases, and confirms Conjecture 3 of Ding and Tang (arXiv:1903.07375).

Significance. The explicit algebraic constructions and parameter formulas, together with the confirmation of an external conjecture, supply new infinite families of 2-designs that can be directly used in coding theory and finite geometry. The fact that the parameters are derived rather than fitted adds concrete value.

minor comments (3)

The abstract states that parameters are 'explicitly determined,' but the manuscript should include a short table (perhaps in §4 or §5) that lists the resulting (v, k, λ) triples for the main families so readers can compare them immediately with known designs.
Notation for the quadratic functions (e.g., the precise form of f(x) = ax² + bx + c and the field F_q) should be fixed once at the beginning of §2 and used consistently; occasional redefinition of the same symbols appears in later sections.
The proof that every pair appears in exactly λ blocks relies on direct evaluation of the number of solutions to certain equations over F_q. A brief remark on why the quadratic degree is essential (or why the argument fails for higher degrees) would clarify the scope of the method.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives infinite families of 2-designs directly from the image sets of quadratic functions over finite fields, computing parameters via explicit finite-field calculations for the listed families and confirming an external conjecture from Ding and Tang. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the design property is asserted only for the specific quadratics considered, with no renaming of known results or smuggling of ansatzes. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about finite fields and quadratic polynomials; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Finite fields admit quadratic functions whose image sets satisfy the 2-design balance condition for the families considered.
Invoked when asserting that the image sets form 2-designs.

pith-pipeline@v0.9.0 · 5646 in / 1179 out tokens · 15860 ms · 2026-05-24T21:33:47.584873+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the incidence structure D(f(x),k) := (GF(q),B(f(x),k)) is a 2-(q,k,λ) design … λ = k(k−1)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

GA(GF(q)) is 2-homogeneous … Theorem 1 [Beth–Jungnickel–Lenz]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 5 internal anchors

[1]

Ding and C

C. Ding and C. Li. Inﬁnite families of 2-designs and 3-des igns from linear codes, Discrete Mathematics, 340: 2415-2431, 2017

work page 2017
[2]

Ding and C

C. Ding and C. Tang. Combinatorial t-designs from special polynomials, arXiv preprint arXiv: 1 903.07375, 2019

work page 2019
[3]

C. Ding. Inﬁnite families of 3-designs from a type of ﬁve- weight code. Des. Codes Cryptogr., 86(3):703-719, 2018

work page 2018
[4]

C. Ding. Designs from linear codes. World Scientiﬁc, 201 8

work page
[5]

C. Ding. Codes from difference sets. World Scientiﬁc, 20 15

work page
[6]

Infinite families of 3-designs from APN functions

C. Tang, Inﬁnite families of 3-designs from APN function s, arXiv:1904.04071, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1904
[7]

C. Tang, C. Ding and M. Xiong, Steiner systems S(2,4,3m− 1 2 ) and 2-designs from ternary linear codes of length 3m− 1 2 . Designs, Codes and Cryptography, 1-19, 2019

work page 2019
[8]

Infinite families of 2-designs from GA_1(q) actions

H. Liu and C. Ding. Inﬁnite families of 2-designs from GA 1(q) actions. arXiv preprint arXiv:1707.02003, 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017
[9]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambri dge University Press, Cambridge, 1999

work page 1999
[10]

M. S. Shrikhande, Quasi-symmetric designs, in: C. J. Co lbourn, and J. H. Dinitz, (Eds.), Handbook of Combinato- rial Designs, 2nd Edition, CRC Press, New Y ork, pp. 102-110, 2007

work page 2007
[11]

X. Du, R. Wang, C. Tang and Q. Wang, Inﬁnite families of 2- designs from two classes of binary cyclic codes with three nonzeros, arXiv:1903.08153v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1903
[12]

X. Du, R. Wang, C. Tang and Q. Wang, Inﬁnite families of 2- designs from two classes of linear codes, arXiv:1903.07459v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1903
[13]

X. Du, R. Wang and C. Fan, Inﬁnite families of 2-designs f rom a class of cyclic codes with two non-zeros, arXiv:1904.04242v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1904
[14]

MacWilliams and N.J.A

F.J. MacWilliams and N.J.A. Sloane. The theory of erro- correcting codes. North-Holland, Amsterdam, 1977

work page 1977
[15]

Electron

Reid C., Rosa A.: Steiner systems S(2,4)-a survey. Electron. J. Comb.DS18, 2010

work page 2010
[16]

Finite Fields Appl

Bluher, A.W.: On xq+1 + ax + b. Finite Fields Appl. 10(3), 285-305, 2004

work page 2004
[17]

E. F. Assmus Jr., J. D. Key, Designs and Their Codes, Camb ridge University Press, Cambridge, 1992

work page 1992
[18]

G. T. Kennedy and V . Pless. A coding-theoretic approach to extending designs. Discrete Math., 142(1-3):155-168, 1995

work page 1995
[19]

Kim and V

J.-L. Kim and V . Pless. Designs in additive codes over GF (4). Des. Codes Cryptogr, 30(2):187C199, 2003

work page 2003
[20]

V . D. Tonchev. Codes and designs. In V . Pless andW. C. Huf fman, editors, Handbook of coding theory, V ol. I, II, pp. 1229-1267. North-Holland, Amsterdam, 1998

work page 1998
[21]

V . D. Tonchev. Codes. In C. J. Colbourn and J. H. Dinitz, e ditors, Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton), pages xxii+ 984. Chapman-Hall/CRC, Boca Raton, FL, second edition, 2007. 13

work page 2007

[1] [1]

Ding and C

C. Ding and C. Li. Inﬁnite families of 2-designs and 3-des igns from linear codes, Discrete Mathematics, 340: 2415-2431, 2017

work page 2017

[2] [2]

Ding and C

C. Ding and C. Tang. Combinatorial t-designs from special polynomials, arXiv preprint arXiv: 1 903.07375, 2019

work page 2019

[3] [3]

C. Ding. Inﬁnite families of 3-designs from a type of ﬁve- weight code. Des. Codes Cryptogr., 86(3):703-719, 2018

work page 2018

[4] [4]

C. Ding. Designs from linear codes. World Scientiﬁc, 201 8

work page

[5] [5]

C. Ding. Codes from difference sets. World Scientiﬁc, 20 15

work page

[6] [6]

Infinite families of 3-designs from APN functions

C. Tang, Inﬁnite families of 3-designs from APN function s, arXiv:1904.04071, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1904

[7] [7]

C. Tang, C. Ding and M. Xiong, Steiner systems S(2,4,3m− 1 2 ) and 2-designs from ternary linear codes of length 3m− 1 2 . Designs, Codes and Cryptography, 1-19, 2019

work page 2019

[8] [8]

Infinite families of 2-designs from GA_1(q) actions

H. Liu and C. Ding. Inﬁnite families of 2-designs from GA 1(q) actions. arXiv preprint arXiv:1707.02003, 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[9] [9]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambri dge University Press, Cambridge, 1999

work page 1999

[10] [10]

M. S. Shrikhande, Quasi-symmetric designs, in: C. J. Co lbourn, and J. H. Dinitz, (Eds.), Handbook of Combinato- rial Designs, 2nd Edition, CRC Press, New Y ork, pp. 102-110, 2007

work page 2007

[11] [11]

X. Du, R. Wang, C. Tang and Q. Wang, Inﬁnite families of 2- designs from two classes of binary cyclic codes with three nonzeros, arXiv:1903.08153v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1903

[12] [12]

X. Du, R. Wang, C. Tang and Q. Wang, Inﬁnite families of 2- designs from two classes of linear codes, arXiv:1903.07459v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1903

[13] [13]

X. Du, R. Wang and C. Fan, Inﬁnite families of 2-designs f rom a class of cyclic codes with two non-zeros, arXiv:1904.04242v1, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1904

[14] [14]

MacWilliams and N.J.A

F.J. MacWilliams and N.J.A. Sloane. The theory of erro- correcting codes. North-Holland, Amsterdam, 1977

work page 1977

[15] [15]

Electron

Reid C., Rosa A.: Steiner systems S(2,4)-a survey. Electron. J. Comb.DS18, 2010

work page 2010

[16] [16]

Finite Fields Appl

Bluher, A.W.: On xq+1 + ax + b. Finite Fields Appl. 10(3), 285-305, 2004

work page 2004

[17] [17]

E. F. Assmus Jr., J. D. Key, Designs and Their Codes, Camb ridge University Press, Cambridge, 1992

work page 1992

[18] [18]

G. T. Kennedy and V . Pless. A coding-theoretic approach to extending designs. Discrete Math., 142(1-3):155-168, 1995

work page 1995

[19] [19]

Kim and V

J.-L. Kim and V . Pless. Designs in additive codes over GF (4). Des. Codes Cryptogr, 30(2):187C199, 2003

work page 2003

[20] [20]

V . D. Tonchev. Codes and designs. In V . Pless andW. C. Huf fman, editors, Handbook of coding theory, V ol. I, II, pp. 1229-1267. North-Holland, Amsterdam, 1998

work page 1998

[21] [21]

V . D. Tonchev. Codes. In C. J. Colbourn and J. H. Dinitz, e ditors, Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton), pages xxii+ 984. Chapman-Hall/CRC, Boca Raton, FL, second edition, 2007. 13

work page 2007