Some Zero-Difference Functions Over $\mathbb{Z}_n$ Using Cyclotomies

Zongxiang Yi

REVIEW 3 major objections 3 minor 19 references

Some Zero-Difference Functions Over $\mathbb{Z}_n$ Using Cyclotomies

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

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Pith's one-line read The paper proposes a generic construction: for any finite ring and any subgroup of its multiplicative group, the coset index function is a zero-difference function.

desk verdict Theorem 2.3 is a clean, useful generalization, but Section 3's claimed families are not proven as written: the key count in Theorem 3.9 is demonstrably false as stated, and the other families rely on sketchy 'similar' proofs. read the letter →

arxiv 1908.09463 v1 pith:PRC3FYO4 submitted 2019-08-19 math.CO

Zongxiang Yi This is my paper · ORCID

classification math.CO MSC 05B1011T22

keywords zero-differencefunctioncosetindexcyclotomicclassesresidueclassringconstantcompositioncodedifferencesystemofsetsfrequency-hoppingsequencebalancedfunctions

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 aims to show that a single construction covers many zero-difference functions: take any finite ring $R$ and any subgroup $G$ of its multiplicative group, then label every element by the coset of $G$ it lies in. The resulting coset index function is always a zero-difference function, meaning that for every nonzero shift $a$, the number of inputs $x$ with $f(x+a)=f(x)$ takes values in a small prescribed set $S$. This reduces the problem of finding zero-difference functions to the problem of counting solutions of the linear equations $x(g-1)=a$ for $g\in G$. The paper applies the method on cyclic rings $\mathbb{Z}_{p^k}$ and on some composite moduli, producing families with $|S|=2$ or $|S|=3$; such functions are used as ingredients for constant composition codes, difference systems of sets, and frequency-hopping sequences.

What carries the argument

The central object is the coset index function $f_G$, which assigns to each ring element the coset $rG$ of the multiplicative subgroup $G$ within the partition $D_G=\{rG: r\in R\}$. The mechanism of the argument is the identity $$\{x\in R: g_G(x+a)=g_G(x)\}=\bigcup_{g\in G}\{x\in R: x(g-1)=a\},$$ which converts the defining difference condition of a zero-difference function into a union of linear equations. The parameters $m$ and $S$ are then read off from coset-size multiplicities $M(G,a)$ and from solution counts $N(G,a)$, with Lemma 3.2 supplying the number of solutions to a linear congruence $ax\equiv b\pmod n$.

What would settle it

For the $n=mp$ family of Theorem 3.9, take $n=12$, $m=4$, $p=3$, $s=2$, $t=1$, $g=2$, so $e=5$ and $G=\{1,5\}$; solving $(g-1)x\equiv 1\pmod{12}$ for $g=5$ gives $4x\equiv 1$, which has no solution, and $g=1$ contributes none, so $N(G,1)=0$, whereas the theorem asserts $m(s-1)=4$; recomputing this count settles whether that family's stated parameters are correct.

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Extended reading notes

Core claim

The central discovery is that every multiplicative subgroup of a finite ring carries a canonical zero-difference function. For a ring $R$ of order $n$ and a subgroup $G$ of $(R,\times)$, the coset index function $f_G(x)=h_G(xG)$ maps $R$ to $\mathbb{Z}_m$, where the cosets $rG$ partition $R$ and $h_G$ encodes them as integers. Theorem 2.3 states that $f_G$ is an $(n,m,S)$ zero-difference function with $m=\sum_{a\in d(G)} M(G,a)/a$ and $S=\{N(G,a): a\in R\setminus\{0\}\}$, where $N(G,a)$ counts the elements $x$ satisfying $x(g-1)=a$ for some $g\in G$. The proof identifies the set of $x$ with $f_G(x+a)=f_G(x)$ exactly with the union of solution sets of those linear equations, so the entire burden shifts to counting. Specializing to $R=\mathbb{Z}_{p^k}$ with generators such as $p-1$ or $2^{k-1}-1$ yields concrete families, for example a $(p^2,p,\{p,p^2-p+1\})$ zero-difference function for every odd prime $p$.

Load-bearing premise

The advertised parameter sets rest on the asserted solution counts $N(G,\alpha)$ for the four families whose proofs are only sketched; if any of those counts is wrong, the corresponding $(n,m,S)$ parameters are not established.

Editorial extensions

If this is right

Every choice of a multiplicative subgroup $G$ of a finite ring yields an explicit zero-difference function, so the construction problem is reduced to computing coset sizes and solution counts.
When $G$ satisfies $(G-1)\setminus\{0\}\subset R^{\times}$, the construction specializes to a zero-difference balanced function with parameters $(n,(n-1)/|G|+1,|G|-1)$, recovering earlier cyclotomic constructions as special cases.
On $\mathbb{Z}_{p^2}$, taking $G=\langle p-1\rangle$ gives a $(p^2,p,\{p,p^2-p+1\})$ zero-difference function for every odd prime $p$.
On $\mathbb{Z}_{2^k}$, taking $G=\langle 2^{k-1}-1\rangle$ gives a $(2^k,2^{k-1}+1,\{0,2\})$ zero-difference function, a family whose difference set $S$ is as small as possible.
The method supplies uniform parameter families for prime powers and for composite moduli of the forms $mp$ and $p_1p_2$, rather than requiring a new construction for each modulus.

Reading between the lines

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

The proof of Theorem 2.3 does not use commutativity in its main step, so the same coset-index construction could be tested on noncommutative rings or products of finite rings, where new parameter triples may arise.
Because the parameter $S$ is fully determined by the function $N(G,a)$, one can enumerate small cyclic subgroups computationally to discover new $(n,m,S)$ triples without additional theory.
The paper's conclusion that these zero-difference functions do not yield optimal codes suggests their likely value is flexibility: small sets $S$ with controlled averages could serve as near-optimal building blocks once a matching bound is formulated for the target application.

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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

The generic method is the real contribution and it is sound. Theorem 2.3 says that any multiplicative subgroup G of a ring R gives a ZDF via the coset index function, with the S-set simply the collection of sizes N(G,a). The proof is short and correct. This genuinely unifies and extends the earlier ZDBF constructions from [16], [2], [8], and [17], and it is honest about the lineage. The paper also deserves credit for stating plainly at the end that the resulting ZDFs do not yield optimal codes; that keeps the significance claim honest.

The trouble is in Section 3, where the new families live. Theorem 3.5 (n = p^2, G = ) is proven in full detail and the counting checks out as far as I can tell. That is a legitimate new family. But Theorems 3.6, 3.7, 3.9, and 3.10 are all justified by 'the proof is similar with that of Theorem 3.5.' That would be fine if the similarity were transparent, but it is not, and in Theorem 3.9 the written proof is not just incomplete—it is wrong.

The stress-test note is correct. In Theorem 3.9, the proof's item (4) claims N(G,α) = 0 for p|α and m(s−1) for p∤α. Take n=12, m=4, p=3, s=2, G={1,5}. For α=1, p∤α, the claimed value is 4, but there are no solutions: g=1 gives 0·x≡1, and g=5 gives 4x≡1 (mod 12), which has no solution because gcd(4,12)=4 does not divide 1. So N(G,1)=0. The theorem's final S-set is still {0,4}, so the statement may survive, but the proof as written does not establish it. The correct count appears to be m(s−1) only when m divides α, not whenever p∤α. That is a different condition, and it needs to be stated and proven.

Given that Theorem 2.3 defines S exactly as the set of N(G,a) values, any unproven or incorrect count leaves the central construction unverified. The same applies to Theorems 3.6, 3.7, and 3.10, where the analogous counts are merely asserted. These could very well be correct—the p^2 case suggests the pattern is real—but the text as it stands does not demonstrate them.

Who is this for? Researchers working on zero-difference functions and their applications. They will want the new parameters, but they should not rely on the Section 3 proofs without doing the counting themselves. The paper is worth a serious referee: the method is solid, the p^2 family is a real new result, and the remaining families are plausible. But it needs a proper revision with full proofs and corrected statements before publication.

Referee Report

3 major / 3 minor

Summary. The paper proposes a generic construction of zero-difference functions (ZDFs) on finite rings via coset index functions induced by multiplicative subgroups, culminating in Theorem 2.3, which expresses the parameters of the resulting ZDF in terms of solution counts N(G,a) of linear equations x(g-1)=a. The paper then applies this method to cyclic rings Z_n, presenting families for n=4, n=2^k, n=p^2, n=p^k, n=mp, and n=p1p2, with parameter sets of size two or three, and summarizes the results in Table 2.

Significance. The generic method is natural, and the proof of Theorem 2.3 is sound; Corollary 2.4 also gives a clean connection to previous zero-difference balanced function constructions without requiring commutativity. The fully proved cases n=2^k and n=p^2 (Theorems 3.4 and 3.5) are concrete and correct. If the remaining families are valid, they would be useful additions to the ZDF literature. However, the later families rest on asserted counting claims that are neither derived nor, in two cases, correct as stated, so the central advertised contributions are not established by the text as written.

major comments (3)

[Theorem 3.9, item (4)] The asserted evaluation of N(G,alpha) is false. For n=12, m=4, p=3, s=2, t=1, G=<5>={1,5}, the proof claims N(G,1)=m(s-1)=4, but the congruence (5-1)x=4x=1 (mod 12) has no solution, so N(G,1)=0. The correct divisibility condition for the positive value is m|alpha, not p∤alpha: the nonzero counts occur for alpha divisible by m, and the statement as printed is therefore incorrect. The final S set may still be salvageable because values m(s-1) are attained for alphamultiple of m, but the proof must be rewritten with the correct condition and a full derivation.
[Theorem 3.7, item (4)] The asserted formula N(G,alpha)=sum_{j=0}^i phi(p^{k-j}) for p^i||alpha with k-s≤i≤k-1 is also false. Take p=3, k=4, s=2, so n=81 and G=<10>={1+9t : t=0,...,8}. For alpha=9 (i=2), the printed value is phi(81)+phi(27)+phi(9)=54+18+6=78, but the equations 9t x≡9 (mod 81) for t=1,...,8 have as their union exactly the 54 residues x with x mod 9 a unit, so N(G,9)=54. The same discrepancy occurs for alpha=27. Although 54 is still in the theorem's declared parameter set S, the written count is wrong, and this further demonstrates that the sketched counting arguments cannot be taken at face value.
[Sections 3.D and 3.E] The proofs of Theorems 3.6, 3.7, 3.9, and 3.10 consist of asserted lists of structural and counting facts with no derivation for item (4), the evaluations of N(G,alpha). Since Theorem 2.3 defines the ZDF parameter S exactly as the set of these counts, the missing computations are load-bearing. The two counterexamples in Theorems 3.7 and 3.9 show that 'the proof is similar with that of Theorem 3.5' is not a reliable substitute for proof in this setting. The authors should provide complete derivations for all four families, or remove any family whose claimed parameters cannot be fully justified.

minor comments (3)

[Section 3.D] The sentence 'We have the Theorem 3.5 as follow' at the start of the n=p^k case should refer to Theorem 3.6, not Theorem 3.5.
[Remark 3.1] The word 'communicative' should be 'commutative'; similarly, the conclusion contains the typo 'Serval' for 'Several'.
[Theorem 3.7, item (2)] The notation p^i||alpha is used to define the cases in item (2), but it is not defined before first use; a sentence explaining that p^i||alpha means p^i is the largest power of p dividing alpha would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: Theorem 2.3 is a self-contained reduction to solving linear congruences, and the Section 3 families explicitly compute those solution counts. The only self-citation, to the author's prior [16], is acknowledged for Corollary 2.4 and is not load-bearing for the new constructions.

full rationale

Theorem 2.3 proves that for the coset-index function f_G the set {x : f_G(x+a)=f_G(x)} equals the union over g in G of {x : x(g-1)=a}, so the ZDF count for a is exactly N(G,a). The theorem then defines S={N(G,a) | a in R\{0}}, which makes the ZDF condition true once the equality is proved. This is a definitional reduction to counting solutions of linear equations, but it is not a circular prediction: no parameter is fitted, and the target set S is not used as an input to compute N. Corollary 2.4 is explicitly described as 'almost the same as Theorem 1 in [16]', a self-citation, but that result is not used to prove the central construction; Theorem 2.3 is proved in the text from Lemma 2.1 and the ring axioms. The concrete families in Section 3 are also computed directly from the counts N(G,alpha) and image sizes |Im(f_G)|, not read back from the desired parameter sets. The manuscript's genuine weaknesses are correctness and completeness: Theorems 3.6, 3.7, 3.9, and 3.10 are justified only by 'the proof is similar with that of Theorem 3.5', and the proof of Theorem 3.9(4) is false as written (for n=12, m=4, G={1,5}, N(G,1)=0 rather than m(s-1)=4). Those are proof gaps, not circular derivation. Hence the construction chain does not reduce to its own inputs, and the circularity score is low; the score reflects only the minor, non-load-bearing self-citation and the definitional character of Theorem 2.3's S.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper depends on standard algebra and number theory facts; no ad hoc assumptions or fitted numbers are introduced. The main fragility is not in the axioms but in the claimed counts N(G,alpha), which are asserted without proof in several theorems.

assumptions (3)

standard math The cosets rG for a multiplicative subgroup G partition the ring R (Lemma 2.1).
Used to define the coset index function in Section 2.
standard math The linear congruence ax=b (mod n) has exactly d solutions when d=gcd(a,n) divides b, and none otherwise (Lemma 3.2).
Used to count solution sets in all applications in Section 3.
standard math Z_p^+ is cyclic for prime p, so generators exist (Theorems 3.9, 3.10).
Used to define the elements e whose powers generate G in Theorems 3.9 and 3.10.

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Pith. "Pith review of Some Zero-Difference Functions Over $\mathbb{Z}_n$ Using Cyclotomies." pith.science (2026). https://pith.science/paper/PRC3FYO4

@misc{pith2026190809463,
  author       = {Pith},
  title        = {Pith review of: Some Zero-Difference Functions Over $\mathbbZ_n$ Using Cyclotomies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRC3FYO4}},
  note         = {Machine review of arXiv:1908.09463}
}

abstract

A generic method to construct zero-difference functions (ZDFs) on algebraic rings is proposed in this paper. Then this method is used over some rings $\mathbb{Z}_{p^k}$, where $p$ is a prime number and $k\ge 2$ is a positive integer, and for some other special rings.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

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

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A generic method to construct zero-difference balanced functions,

Z. Yi, Z. Lin, and L. Ke, “A generic method to construct zero-difference balanced functions,” Cryptography and Communications, vol. 10, no. 4, pp. 591–609, 2018

work page 2018

[2] [2]

A new construction of zero-difference balanced functions and its applications,

H. Cai, X. Zeng, T. Helleseth, X. Tang, and Y . Yang, “A new construction of zero-difference balanced functions and its applications,” IEEE Transactions on Information Theory , vol. 59, no. 8, pp. 5008–5015, 2013

work page 2013

[3] [8]

Three new families of zero-difference balanced functions with applications,

C. Ding, Q. Wang, and M. Xiong, “Three new families of zero-difference balanced functions with applications,” IEEE Transactions on Information Theory , vol. 60, no. 4, pp. 2407–2413, 2014

work page 2014

[4] [17]

Cyclotomic constructions of zero-difference balanced functions with applications,

Z. Zha and L. Hu, “Cyclotomic constructions of zero-difference balanced functions with applications,” IEEE Transactions on Information Theory , vol. 61, no. 3, pp. 1491–1495, 2015

work page 2015

[5] [1]

Partitioned difference families versus zero-difference balanced functions,

M. Buratti and D. Jungnickel, “Partitioned difference families versus zero-difference balanced functions,” Designs, Codes and Cryptography, 2019

work page 2019

[6] [3]

Zero-difference balanced functions with new parameters and their applications,

H. Cai, Z. Zhou, X. Tang, and Y . Miao, “Zero-difference balanced functions with new parameters and their applications,” IEEE Transactions on Information Theory , vol. 63, no. 7, pp. 4379–4387, 2017

work page 2017

[7] [4]

Highly nonlinear mappings,

C. Carlet and C. Ding, “Highly nonlinear mappings,” Journal of complexity , vol. 20, no. 2, pp. 205–244, 2004

work page 2004

[8] [5]

Quadratic zero-difference balanced functions, apn functions and strongly regular graphs,

C. Carlet, G. Gong, and Y . Tan, “Quadratic zero-difference balanced functions, apn functions and strongly regular graphs,” Designs, Codes and Cryptography , pp. 1–26, 2014

work page 2014