Paley type partial difference sets in abelian groups

Zeying Wang

Paley type partial difference sets in abelian groups

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a Paley type partial difference set exists in some abelian group of odd order $v>1$ exactly when $v$ is a prime power congruent to 1 modulo 4, or $v=n^4$ or $9n^4$ for odd $n>1$.

desk verdict Clean, correct proof that completes the 25-year-old existence classification for Paley-type PDSs in abelian groups; a bit narrow, very solid. read the letter →

arxiv 1908.07055 v1 pith:QL6DNLMX submitted 2019-08-19 math.CO

Zeying Wang This is my paper · ORCID

classification math.CO MSC 05B1005E3005C25

keywords partialdifferencesetsPaleytypeabeliangroupsCayleygraphsstronglyregularsub-partialprimepowers

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

This paper closes a long-open existence question: for which odd orders $v>1$ does some abelian group contain a Paley type partial difference set? The author proves that when $v$ is not a prime power, such a set can exist only if $v=n^4$ or $v=9n^4$ for an odd integer $n>1$. Since the classical finite-field construction gives all prime powers congruent to 1 modulo 4, and an earlier construction gives the orders $n^4$ and $9n^4$, the characterization is complete. The upshot is a clean division: either the order is a prime power congruent to 1 modulo 4, or it is a fourth power, possibly multiplied by 9.

What carries the argument

The load-bearing mechanism is the sub-partial-difference-set theorem (Theorem 2.1, quoted from [1]): when a nontrivial regular PDS with square discriminant $\Delta=\delta^2$ is restricted to a subgroup $H$ whose order is coprime to its index and whose index is odd, the intersection $D\cap H$ is again a regular PDS with parameters controlled by $\pi=\gcd(|H|,\delta)$ and an integer $\theta$ chosen so that $(2\theta-1)\pi \le \beta < (2\theta+1)\pi$. The crucial clause says that if $\delta=p^r\pi$ with $p\ge 5$ and $\gcd(p,\pi)=1$, then odd $r$ forces $\theta\equiv (p-1)/2 \pmod{p-1}$. For a Paley type PDS one has $\beta=-1$, so in the chosen subgroup $\theta=0$; the congruence then contradicts odd $r$. This single mechanism extracts the square structure of the group order from the PDS parameters.

What would settle it

Run a complete character-theoretic search for a $(1225, 612, 305, 306)$-partial difference set in each of the four abelian groups of order $5^2 \cdot 7^2$; finding one would refute Theorem 1.2, which predicts none because 1225 is neither a prime power congruent to 1 modulo 4 nor of the form $n^4$ or $9n^4$.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: for an odd integer $v>1$, a Paley type partial difference set (PDS) exists in some abelian group of order $v$ if and only if $v$ is a prime power with $v\equiv 1 \pmod{4}$, or $v=n^4$ or $9n^4$ for odd $n>1$. The new ingredient is the necessity direction for orders that are not prime powers. Given a Paley type PDS in a group of order $p^{2r}u^2$ with $p\ge 5$ and $\gcd(p,u)=1$, the author takes a subgroup $H$ of order $u^2$ and applies a theorem on sub-partial-difference sets to force the auxiliary integer $\theta$ to be $0$; the theorem's congruence condition then rules out odd $r$. Hence every prime $p\ge 5$ appears with exponent divisible by 4, leaving only fourth powers possibly multiplied by 9. Sufficiency comes from the existing finite-field and group constructions.

Load-bearing premise

The argument depends on a previously established theorem about how a partial difference set restricts to a subgroup, including a precise congruence condition on an auxiliary integer; if that theorem is wrong or does not apply to the chosen subgroup, the proof that rules out non-prime-power orders collapses.

Editorial extensions

If this is right

The existence question for Paley type partial difference sets in abelian groups of odd order is fully answered, with no order left undecided.
Any order of the form $p^2 q^2$ with distinct primes $p,q$ at least 5 cannot support a Paley type PDS; only fourth-power exponents, with the single possible factor 9, survive.
All admissible non-prime-power orders already have explicit constructions, so the necessary condition is sharp.
For prime-power orders, the classical finite-field construction is the only source needed.

Reading between the lines

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

The classification also settles which abelian groups can carry strongly regular Cayley graphs with $\lambda-\mu=-1$, since a Paley type PDS is exactly the Cayley subset that produces such a graph; the paper states the result only in PDS language.
The necessity proof uses only the order of the group, not its abelian structure, so the finer question of which particular abelian groups of order $n^4$ or $9n^4$ admit such a set remains open; the known construction provides at least one group for each admissible order.
The exceptional factor 9 points to the prime 3 as the only prime whose square can appear outside the fourth-power part; working out the $p=3$ case of the congruence machinery might show whether this exception is forced or merely an artifact of the construction.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a clean short proof that completes the classification of orders for Paley-type PDSs in abelian groups. The main theorem is exactly the if-and-only-if statement: prime power v ≡ 1 mod 4, or v = n^4 or 9n^4 for odd n > 1. The necessary side was the open part; the paper proves Theorem 2.2, that for p ≥ 5 the exponent r in p^{2r}u^2 must be even, and derives the order restriction.

What I like: the proof is short and transparent. It uses the Arasu–Jungnickel–Ma–Pott theorem (2.1) to pull the sub-PDS D1 back to a subgroup of order u^2, computes β1 = −1 and k1 = (u^2−1)/2, and then uses the congruence clause to force r even. I checked the calculation: π = gcd(u^2, p^r u) = u, the interval for θ contains −1 only for θ = 0, and the congruence for odd r gives θ ≡ (p−1)/2 mod p−1, which contradicts θ = 0. So the load-bearing step holds. The reduction from the corollary to the main theorem is fine, and Polhill’s constructions are cited for the converse. The paper also openly acknowledges that Leung and Ma had listed a similar lemma without proof; that’s the right thing to do, and it keeps the novelty claim honest.

Soft spots, all minor: Theorem 2.1 is quoted without proof. That is fine given it is published, but the “Moreover” clause is doing real work, so a reader might want the statement checked against the original. There is a small typo in the proof of Theorem 2.2: it says “D ≠ H \ {e}” where it should say “D1 ≠ H \ {e}”. The claim is true with D1, and the printed statement is meaningless in context. Also the paper could have noted explicitly that primes p = 3 cause no problem in Corollary 2.3; the conclusion still follows because the constraint is only on p ≥ 5, and an odd exponent on 3 just produces the 9n^4 factor. Not a gap, but a line of explanation would help.

Bottom line: the result is correct, the prose is clear, and the history is handled honestly. It is a solid subfield contribution, not a revolution. Anyone working on partial difference sets or strongly regular Cayley graphs will want this for the classification statement. I would send it to a serious combinatorics journal like JCTB or Designs, Codes and Cryptography. It deserves peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies Paley type partial difference sets in abelian groups, i.e. regular PDSs with parameters (v, (v-1)/2, (v-5)/4, (v-1)/4). It first recalls a theorem of Ma stating that if the discriminant Delta is not a square, then v must be a prime power congruent to 1 modulo 4. The main work concerns square v. Using a sub-PDS theorem of Arasu, Jungnickel, Ma, and Pott (Theorem 2.1), the paper proves Theorem 2.2: no Paley type PDS can exist in an abelian group of order p^{2r}u^2 with p >= 5, gcd(p,u)=1, and u>1 when r is odd. Applying this to each prime divisor p_i >= 5 of a square non-prime-power order forces every such exponent to be a multiple of 4, so the order is n^4 or 9n^4 for odd n>1 (Corollary 2.3). Combined with Polhill's 2010 constructions and the classical Paley construction over finite fields, this yields the complete characterization in Theorem 1.2: for odd v>1, a Paley type PDS exists in some abelian group of order v if and only if v is a prime power congruent to 1 modulo 4, or v = n^4 or 9n^4 for odd n>1.

Significance. If correct, this result completely answers the motivating existence question for odd orders, the natural completion of a line of work by Ma, Arasu-Jungnickel-Ma-Pott, Polhill, and the author. The proof is short and transparent: the main technical input is the quoted Theorem 2.1, and the application to a subgroup of order u^2 is clean and checks out. The paper gives explicit credit to prior work and, in the acknowledgement, appropriately notes the related Lemma 2.4 of Leung and Ma. The necessary condition derived here is sharp because matching constructions were already known, making this a meaningful and publishable contribution.

minor comments (3)

[§2, Corollary 2.3] In the proof of Corollary 2.3, the quantity u should be defined as u = sqrt(|G|/p_i^{2t_i}), not as |G|/p_i^{2t_i}; the group order is p_i^{2t_i} u^2 in Theorem 2.2, and |G|/p_i^{2t_i} is a square because all other exponents are even.
[§2, Theorem 2.2 proof] In the proof of Theorem 2.2, the sentence "Also D \neq H \setminus {e}" should read "Also D_1 \neq H \setminus {e}", since D_1 is the intersection of D with H and the condition in Theorem 2.1 concerns D_1.
[Throughout] There are several typographical errors that should be corrected: "Athough" for "Although", "Thorem" for "Theorem", and "relative prime" for "relatively prime".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a direct deduction from external theorems and constructions; the sole self-citation is unused.

full rationale

Walking the derivation chain: Theorem 1.2 is an iff. The forward direction splits by Δ = |G|. If |G| is not a square, Ma's Theorem 1.1 (external, [4]) forces |G| = p^{2s+1}, p≡1 mod 4. If |G| is a square and not a prime power, the paper forms H of order u^2 for each p_i ≥ 5 and invokes Theorem 2.1 from [1] (external) with π = gcd(|H|, δ) = u, β = -1, θ = 0. The 'Moreover' clause then excludes odd r by θ ≡ (p-1)/2 mod p-1, contradiction. This is a direct instantiation of an independent external theorem; no hypothesis of Theorem 2.1 contains the desired conclusion, and the algebra quoted is explicit. The backward direction uses the classical Paley construction and Polhill's 2010 construction [9], again external. The only self-citation, [10], is mentioned as prior partial work and is never cited in the proof. The acknowledgement that Leung and Ma listed a similar lemma is not load-bearing circularity. Therefore I find no step in which a claimed result is equivalent by construction to an input, a fitted parameter is relabeled as a prediction, or a load-bearing uniqueness claim is imported solely from the authors' own prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The proof relies on established theorems (Ma, AJMP, Polhill, Paley) and standard group theory; these are listed as domain assumptions.

assumptions (5)

domain assumption Theorem 2.1 (Arasu-Jungnickel-Ma-Pott sub-PDS theorem) is correct, including the 'Moreover' congruence condition.
Invoked in the proof of Theorem 2.2 to obtain the parameters of D1 and the contradiction on θ.
domain assumption Theorem 1.1 (Ma 1984) is correct.
Used in the introduction to restrict attention to square orders.
domain assumption Theorem 2.4 (Polhill 2010) is correct.
Provides existence of Paley type PDS for orders n^4 and 9n^4.
standard math Standard Hall subgroup existence in abelian groups.
Needed to choose H of order u^2 in Theorem 2.2.
domain assumption Paley's construction for prime powers q ≡ 1 (mod 4).
Used for the prime power sufficiency direction.

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Cite this review

Pith. "Pith review of Paley type partial difference sets in abelian groups." pith.science (2026). https://pith.science/paper/QL6DNLMX

@misc{pith2026190807055,
  author       = {Pith},
  title        = {Pith review of: Paley type partial difference sets in abelian groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QL6DNLMX}},
  note         = {Machine review of arXiv:1908.07055}
}

abstract

Partial difference sets with parameters $(v,k,\lambda,\mu)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference set in an abelian group $G$ of an order not a prime power, then $|G|=n^4$ or $9n^4$, where $n>1$ is an odd integer. In 2010, Polhill \cite{Polhill} constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using non-zero squares of a finite field, we completely answer the following question: "For which odd positive integer $v > 1$, can we find a Paley type partial difference set in an abelian group of order $v$?"

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Arasu, D

K.T. Arasu, D. Jungnickel, S.L. Ma and A. Pott, Strongly r egular Cayley graphs with λ − µ = −1, Journal of Combinatorial Theory , Series A, 67, 116-125 (1994)

work page 1994
[2]

Davis, Partial diﬀerence sets in p-groups

J. Davis, Partial diﬀerence sets in p-groups. Arch. Math. (Basel) , 63, no. 2, 103–110, (1994)

work page 1994
[3]

Leung, S.L

K.H. Leung, S.L. Ma, Partial diﬀerence sets with Paley par ameters, Bull. London Math. Soc. 27, no. 6, 553–564 (1995)

work page 1995
[4]

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

work page 1984
[5]

S. L. Ma, On subsets of partial diﬀerence sets, Discrete Mathematics 125, 263- 272 (1994)

work page 1994
[6]

Ma, A survey of partial diﬀerence sets, Designs, Codes, Cryptogr

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

work page 1994
[7]

Paley, On orthogonal matrices, J

R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys. 12, 311-320 (1933)

work page 1933
[8]

Polhill, Paley partial diﬀerence sets in non- p-groups, Designs, Codes, Cryp- togr

J. Polhill, Paley partial diﬀerence sets in non- p-groups, Designs, Codes, Cryp- togr. 52, 163-169 (2009) 4

work page 2009

Show all 10 references

[9]

Polhill, Paley partial diﬀerence sets in groups of orde r n4 and 9n4 for any odd n > 1, Journal of Combinatorial Theory , Series A 117, 1027–1036 (2010)

J. Polhill, Paley partial diﬀerence sets in groups of orde r n4 and 9n4 for any odd n > 1, Journal of Combinatorial Theory , Series A 117, 1027–1036 (2010)

2010
[10]

Wang, New necessary conditions on Paley type partial diﬀerence sets in Abelian groups, Journal of Combinatorial Designs 27, 415-419 ( 2019) 5

Z. Wang, New necessary conditions on Paley type partial diﬀerence sets in Abelian groups, Journal of Combinatorial Designs 27, 415-419 ( 2019) 5

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Arasu, D

K.T. Arasu, D. Jungnickel, S.L. Ma and A. Pott, Strongly r egular Cayley graphs with λ − µ = −1, Journal of Combinatorial Theory , Series A, 67, 116-125 (1994)

work page 1994

[2] [2]

Davis, Partial diﬀerence sets in p-groups

J. Davis, Partial diﬀerence sets in p-groups. Arch. Math. (Basel) , 63, no. 2, 103–110, (1994)

work page 1994

[3] [3]

Leung, S.L

K.H. Leung, S.L. Ma, Partial diﬀerence sets with Paley par ameters, Bull. London Math. Soc. 27, no. 6, 553–564 (1995)

work page 1995

[4] [4]

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

work page 1984

[5] [5]

S. L. Ma, On subsets of partial diﬀerence sets, Discrete Mathematics 125, 263- 272 (1994)

work page 1994

[6] [6]

Ma, A survey of partial diﬀerence sets, Designs, Codes, Cryptogr

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

work page 1994

[7] [7]

Paley, On orthogonal matrices, J

R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys. 12, 311-320 (1933)

work page 1933

[8] [8]

Polhill, Paley partial diﬀerence sets in non- p-groups, Designs, Codes, Cryp- togr

J. Polhill, Paley partial diﬀerence sets in non- p-groups, Designs, Codes, Cryp- togr. 52, 163-169 (2009) 4

work page 2009