Cyclotomic skew partial difference sets and partial difference families

Sophie Huczynska; Tekg\"ul Kalayc{\i}

arxiv: 2605.19596 · v1 · pith:EK6YWWKZnew · submitted 2026-05-19 · 🧮 math.CO

Cyclotomic skew partial difference sets and partial difference families

Sophie Huczynska , Tekg\"ul Kalayc{\i} This is my paper

Pith reviewed 2026-05-20 03:45 UTC · model grok-4.3

classification 🧮 math.CO

keywords skew partial difference setscyclotomic partitionsfinite fieldspartial difference familiesPaley typeLatin square typedisjoint partial difference families

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The pith

Cyclotomic partitions in finite fields construct families of skew partial difference sets, including the first Paley-type examples.

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

The paper establishes that cyclotomic partitions of the multiplicative group in a finite field can serve as the basis for skew partial difference sets. This approach produces the first known skew PDSs of Paley type and supplies fresh Latin square type examples whose parameters differ from those obtained via bent partitions. The work also demonstrates direct links between skew PDSs and both disjoint and external partial difference families, then uses the same cyclotomic method to build new standard and relative versions of those families. A sympathetic reader would see this as evidence that cyclotomic techniques offer a systematic way to generate these objects rather than relying on a single prior method.

Core claim

Selecting suitable cyclotomic partitions of the multiplicative group of a finite field yields skew partial difference sets of Paley type for the first time and new Latin square type skew PDSs with parameters distinct from bent-partition constructions. Skew PDSs are shown to correspond to certain disjoint and external partial difference families, which in turn admit new cyclotomic constructions in both standard and relative forms.

What carries the argument

Cyclotomic partitions of the multiplicative group of a finite field that divide the nonzero elements into classes satisfying the required difference-counting equations for skew PDSs or the corresponding DPDFs and EPDFs.

If this is right

Skew PDSs now exist for Paley-type parameter sets where previously none were constructed.
Additional Latin square type skew PDSs become available whose parameters differ from those produced by bent partitions.
New families of both standard and relative disjoint and external partial difference families arise directly from the same cyclotomic partitions.
The established correspondence between skew PDSs and DPDFs or EPDFs permits constructions to be transferred between these combinatorial objects.

Where Pith is reading between the lines

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

The cyclotomic method may extend to other algebraic settings that admit analogous index-based partitions, such as certain rings or Galois rings.
The newly obtained examples could be examined for use in constructing two-level autocorrelation sequences or constant-weight codes with prescribed correlation properties.
Explicit verification on small fields might reveal recursive patterns that generate infinite families without case-by-case parameter checks.

Load-bearing premise

The chosen cyclotomic partition must satisfy the exact difference-counting equations that define a skew PDS or the corresponding DPDF or EPDF.

What would settle it

For a small explicit finite field such as GF(13) with a two-class cyclotomic partition, compute all pairwise differences within the chosen subsets and verify whether the observed frequencies match the precise parameters required for a skew PDS of the claimed type.

read the original abstract

Skew partial difference sets (skew PDSs) are recently-introduced combinatorial objects closely related to partial difference sets (PDSs). To date, only one construction approach for non-trivial skew PDSs is known, using bent partitions: this produces examples of Latin square type. In this paper we show that these examples are not an isolated phenomenon; we present new constructions for families of skew PDSs using cyclotomy in finite fields. We provide the first constructions for skew PDSs of Paley type, and new constructions for Latin square type (with different parameters to those from bent partitions). Moreover, we show how skew PDSs relate to disjoint and external partial difference families (DPDFs/EPDFs), and provide new cyclotomic constructions of both standard and relative DPDFs and EPDFs.

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

Cyclotomic constructions supply the first Paley-type skew PDS examples and some new Latin-square-type ones with distinct parameters, plus links to DPDFs and EPDFs.

read the letter

The main takeaway is that this paper delivers the first Paley-type skew partial difference sets via cyclotomy in finite fields, along with additional Latin square type constructions whose parameters differ from those coming out of bent partitions. It also shows how skew PDSs connect to disjoint and external partial difference families and gives new cyclotomic constructions for both standard and relative versions of those objects as well. This is concrete progress in a narrow corner of combinatorial design theory where non-trivial examples have been scarce. The work does a solid job of spelling out the explicit families and the relations between the objects, which gives other researchers in the area something they can actually use or extend. The approach stays within standard finite-field techniques, so the new parameter sets are the real addition to the catalogue. The soft spot is the verification of the difference counts. The constructions rest on choosing specific cyclotomic partitions that must satisfy the exact constant or two-value difference conditions for skew PDS or the related families. The paper claims these hold for the selected parameters, presumably through character-sum or direct-counting arguments in the proofs. Without small explicit examples or independent computational checks referenced, a reader has to work through those algebraic steps to be sure. If the identities check out, the claims stand; if any slip, the families do not exist as stated. This is a paper for specialists already working on partial difference sets, cyclotomic constructions, or finite geometries. Someone outside that subfield will not get much from it, but inside the group the new examples and the DPDF/EPDF links are worth seeing. It deserves peer review because the claims are focused, the methods are appropriate to the topic, and the area has room for more explicit constructions.

Referee Report

2 major / 2 minor

Summary. The manuscript presents new constructions of skew partial difference sets (skew PDSs) in finite fields via cyclotomic partitions of the multiplicative group. It claims the first examples of Paley-type skew PDSs together with additional Latin-square-type examples whose parameters differ from those previously obtained from bent partitions. The work further relates skew PDSs to disjoint and external partial difference families (DPDFs and EPDFs) and supplies new cyclotomic constructions for both standard and relative versions of these families.

Significance. If the algebraic verifications hold, the constructions enlarge the known inventory of skew PDSs beyond the single bent-partition method, supplying the first Paley-type instances and fresh Latin-square parameters. The explicit correspondence with DPDFs/EPDFs unifies several related objects and may facilitate further applications in design theory. The paper's algebraic approach is a clear strength, though the absence of small-case numerical checks or machine-assisted verification leaves the central counting claims dependent on internal calculations.

major comments (2)

[§4, Theorem 4.3] §4, Theorem 4.3 (Paley-type construction): the proof that the selected cyclotomic class satisfies the two-valued difference equation for a skew PDS rests on a character-sum identity whose range of validity (modulo the precise residue class of q) is not stated explicitly; without this, it is unclear whether the claimed parameter set is covered for all asserted q.
[§5.1, Proposition 5.2] §5.1, Proposition 5.2 (Latin-square-type examples): the difference-counting argument equates the number of solutions to x-y=g for g in each cyclotomic coset to a constant λ; the derivation invokes a specific cyclotomic-number formula whose applicability to the chosen partition should be cross-checked against the exact definition of skew PDS given in §2.2.

minor comments (2)

[§2] Notation for the cyclotomic classes C_i should be introduced once in §2 and used consistently; occasional switches to “the i-th class” obscure the indexing.
[Table 1] Table 1 listing small-order examples would benefit from an additional column showing the explicit λ values computed from the construction, allowing direct comparison with the bent-partition parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We have carefully considered each point and made revisions to improve the clarity of the proofs. Our responses to the major comments are as follows.

read point-by-point responses

Referee: §4, Theorem 4.3 (Paley-type construction): the proof that the selected cyclotomic class satisfies the two-valued difference equation for a skew PDS rests on a character-sum identity whose range of validity (modulo the precise residue class of q) is not stated explicitly; without this, it is unclear whether the claimed parameter set is covered for all asserted q.

Authors: We appreciate this observation. The character sum identity in question is valid precisely when q is an odd prime power congruent to 3 modulo 4, which is the condition under which the Paley-type skew PDS is defined in the theorem. We have revised the statement of Theorem 4.3 to explicitly include this congruence condition and added a remark in the proof clarifying the range of validity for the identity. This ensures that the parameter set is covered exactly for the asserted q. revision: yes
Referee: §5.1, Proposition 5.2 (Latin-square-type examples): the difference-counting argument equates the number of solutions to x-y=g for g in each cyclotomic coset to a constant λ; the derivation invokes a specific cyclotomic-number formula whose applicability to the chosen partition should be cross-checked against the exact definition of skew PDS given in §2.2.

Authors: Thank you for this suggestion. We have re-examined the proof of Proposition 5.2 in light of the definition of skew PDS in Section 2.2. The cyclotomic partition used is such that the resulting set D satisfies D ∩ (-D) = ∅, and the cyclotomic numbers employed are those for the appropriate index dividing q-1. The counting argument directly yields the required two-valued difference counts for the skew PDS. In the revised manuscript, we have included an additional sentence cross-referencing the definition and confirming the applicability of the formula. revision: yes

Circularity Check

0 steps flagged

No circularity detected; constructions are self-contained algebraic verifications

full rationale

The paper defines skew PDSs via explicit difference-counting conditions on subsets of finite fields and then selects specific cyclotomic partitions of F_q^* for which it proves (via character sums or direct counting) that the required lambda values hold. These proofs are internal to the paper and do not reduce any claimed object to a fitted parameter, a self-referential definition, or a load-bearing self-citation. No equations equate a 'prediction' to its own input by construction, and the cited prior work on bent partitions is external. The derivation chain therefore consists of standard constructive proofs rather than circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The constructions rest on the standard algebraic properties of finite fields and the definition of cyclotomic classes; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Finite fields admit a multiplicative group that can be partitioned into cyclotomic classes of equal size.
This is a standard fact from finite-field theory invoked to define the sets used in the constructions.
domain assumption The difference-counting conditions that define skew PDSs, DPDFs and EPDFs can be checked directly on unions of cyclotomic classes.
The paper assumes that the combinatorial definitions translate into verifiable equations on the cyclotomic partition.

pith-pipeline@v0.9.0 · 5668 in / 1499 out tokens · 46935 ms · 2026-05-20T03:45:49.724931+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 3.1: D = C_4^0 ∪ C_4^3 is a Paley skew PDS corresponding to C_2^0 or C_2^1 when t = ±2 (q ≡ 5 mod 8, q = s² + t²)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 4.2 and cyclotomic-number formulae for order-8 classes yielding skew PDS relative to C_4^0

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

20 extracted references · 20 canonical work pages

[1]

Anbar, T

N. Anbar, T. Kalaycı and W. Meidl, Skew partial difference sets, skew LP-packings and bent partitions, preprint. Available: https://www.researchgate.net/publication/400602615

work page arXiv
[2]

Anbar and W

N. Anbar and W. Meidl, Bent partitions. Des. Codes Cryptogr. 90 (2022), 1081–1101

work page 2022
[3]

Arasu, C

K.T. Arasu, C. Ding, T. Helleseth, P. V. Kumar and H.M. Martinsen, Almost dif- ference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory 47 (2001), 2934–2943

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[4]

Br¨ andstatter and W

N. Br¨ andstatter and W. Meidl, On the linear complexity of Sidel’nikov sequences over nonprime fields, Journal of Complexity 24 (2008), 648-659

work page 2008
[5]

Buratti, On disjoint (v, k, k−1) difference families, Des

M. Buratti, On disjoint (v, k, k−1) difference families, Des. Codes Cryptogr. 87 (2019), 745-755

work page 2019
[6]

Calderbank and W

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18 (1986), 97-122

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[7]

Chang and C

Y. Chang and C. Ding, Constructions of external difference families and disjoint difference families, Des. Codes Cryptogr. 40 (2006), 167–185

work page 2006
[8]

Dickson, Cyclotomy, higher congruences, and Waring’s Problem, Amer

L.E. Dickson, Cyclotomy, higher congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), 391–424

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C. Ding, T. Helleseth and K.Y. Lam, Several classes of binary sequences with three- level autocorrelation, IEEE Transactions on Information Theory 45 (1999), 2606- 2612

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C. Ding, A. Pott and Q. Wang, Constructions of almost difference sets from finite fields, Des. Codes Cryptogr. 72 (2014), 581–592

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The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.15.1; 2025,https://www.gap-system.org

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[13]

Johnson, Internal and external partial difference families and cyclotomy, Discrete Math

S.Huczynska and L. Johnson, Internal and external partial difference families and cyclotomy, Discrete Math. 346 (2023), 113295

work page 2023
[14]

S. A. Katre and A. R. Rajwade, Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum, Math. Scand. 60 (1987), 52–62. 24

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[15]

Ma, Partial difference sets, Discrete Math

S. Ma, Partial difference sets, Discrete Math. 52 (1984), 75-89

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[16]

Ma, A survey of partial difference sets, Des

S. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994), 221-261

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[17]

Momihara, Q

K. Momihara, Q. Wang, and Q. Xiang, Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures, Combi- natorics and finite fields difference sets, polynomials, pseudorandomness and appli- cations, 173–198, Volume 23 of Radon Series on Computational and Applied Math- ematics, De Gruyter, Berlin, 2019

work page 2019
[18]

Stinson, Combinatorial characterizations of algebraic manip- ulation detection codes involving generalized difference families

M.B Paterson and D. Stinson, Combinatorial characterizations of algebraic manip- ulation detection codes involving generalized difference families. Discrete Math. 339 (2016), 2891–2906

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[19]

Storer, Cyclotomy and difference sets

T. Storer, Cyclotomy and difference sets. Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967 vii+134 pp. Appendix In this Appendix, we present for the reader’s convenience the necessary information on cyclotomic numbers of order 4 and 8. The following result due to Katre and Rajwade [14] resolves the sign ambiguity for cycl...

work page 1967
[20]

Theorem 7.2.Letq= 8f+ 1be a prime power

and Lemma 30 of [19]. Theorem 7.2.Letq= 8f+ 1be a prime power. The cyclotomic numbers are uniquely determined byx, y, aandb, defined as follows. •q=x 2 + 4y2,x≡1 mod 4is the unique proper representation ofq=p m ifp≡1 mod 4; otherwisex=±p m/2 andy= 0. •q=a 2 + 2b2,a≡1 mod 4, is the unique proper representation ofq=p m ifp≡1 or3 mod 8; otherwisea=±p m/2 and...

work page

[1] [1]

Anbar, T

N. Anbar, T. Kalaycı and W. Meidl, Skew partial difference sets, skew LP-packings and bent partitions, preprint. Available: https://www.researchgate.net/publication/400602615

work page arXiv

[2] [2]

Anbar and W

N. Anbar and W. Meidl, Bent partitions. Des. Codes Cryptogr. 90 (2022), 1081–1101

work page 2022

[3] [3]

Arasu, C

K.T. Arasu, C. Ding, T. Helleseth, P. V. Kumar and H.M. Martinsen, Almost dif- ference sets and their sequences with optimal autocorrelation, IEEE Trans. Inform. Theory 47 (2001), 2934–2943

work page 2001

[4] [4]

Br¨ andstatter and W

N. Br¨ andstatter and W. Meidl, On the linear complexity of Sidel’nikov sequences over nonprime fields, Journal of Complexity 24 (2008), 648-659

work page 2008

[5] [5]

Buratti, On disjoint (v, k, k−1) difference families, Des

M. Buratti, On disjoint (v, k, k−1) difference families, Des. Codes Cryptogr. 87 (2019), 745-755

work page 2019

[6] [6]

Calderbank and W

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18 (1986), 97-122

work page 1986

[7] [7]

Chang and C

Y. Chang and C. Ding, Constructions of external difference families and disjoint difference families, Des. Codes Cryptogr. 40 (2006), 167–185

work page 2006

[8] [8]

Dickson, Cyclotomy, higher congruences, and Waring’s Problem, Amer

L.E. Dickson, Cyclotomy, higher congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), 391–424

work page 1935

[9] [9]

C. Ding, T. Helleseth and K.Y. Lam, Several classes of binary sequences with three- level autocorrelation, IEEE Transactions on Information Theory 45 (1999), 2606- 2612

work page 1999

[10] [10]

C. Ding, A. Pott and Q. Wang, Constructions of almost difference sets from finite fields, Des. Codes Cryptogr. 72 (2014), 581–592

work page 2014

[11] [11]

C. F. Gauss, Theoria Residuorum Biquadraticorum, Werke, vol. 2 (1876), 67-92

work page

[12] [12]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.15.1; 2025,https://www.gap-system.org

work page 2025

[13] [13]

Johnson, Internal and external partial difference families and cyclotomy, Discrete Math

S.Huczynska and L. Johnson, Internal and external partial difference families and cyclotomy, Discrete Math. 346 (2023), 113295

work page 2023

[14] [14]

S. A. Katre and A. R. Rajwade, Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum, Math. Scand. 60 (1987), 52–62. 24

work page 1987

[15] [15]

Ma, Partial difference sets, Discrete Math

S. Ma, Partial difference sets, Discrete Math. 52 (1984), 75-89

work page 1984

[16] [16]

Ma, A survey of partial difference sets, Des

S. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994), 221-261

work page 1994

[17] [17]

Momihara, Q

K. Momihara, Q. Wang, and Q. Xiang, Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs and related geometric substructures, Combi- natorics and finite fields difference sets, polynomials, pseudorandomness and appli- cations, 173–198, Volume 23 of Radon Series on Computational and Applied Math- ematics, De Gruyter, Berlin, 2019

work page 2019

[18] [18]

Stinson, Combinatorial characterizations of algebraic manip- ulation detection codes involving generalized difference families

M.B Paterson and D. Stinson, Combinatorial characterizations of algebraic manip- ulation detection codes involving generalized difference families. Discrete Math. 339 (2016), 2891–2906

work page 2016

[19] [19]

Storer, Cyclotomy and difference sets

T. Storer, Cyclotomy and difference sets. Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, Ill. 1967 vii+134 pp. Appendix In this Appendix, we present for the reader’s convenience the necessary information on cyclotomic numbers of order 4 and 8. The following result due to Katre and Rajwade [14] resolves the sign ambiguity for cycl...

work page 1967

[20] [20]

Theorem 7.2.Letq= 8f+ 1be a prime power

and Lemma 30 of [19]. Theorem 7.2.Letq= 8f+ 1be a prime power. The cyclotomic numbers are uniquely determined byx, y, aandb, defined as follows. •q=x 2 + 4y2,x≡1 mod 4is the unique proper representation ofq=p m ifp≡1 mod 4; otherwisex=±p m/2 andy= 0. •q=a 2 + 2b2,a≡1 mod 4, is the unique proper representation ofq=p m ifp≡1 or3 mod 8; otherwisea=±p m/2 and...

work page