{"id":"294f7d15-ed76-4184-8bd1-e21885303852","arxiv_id":"1908.09463","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The coset index function induced by a cyclic multiplicative subgroup is shown to be a zero-difference function with small difference sets for many rings Z_n.","lead":"This paper constructs zero-difference functions, maps used in coding theory, by assigning a coset index to each element of the integers modulo n. It offers several new parameter families, but the author notes they do not yield optimal codes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The generic Theorem 2.3 is sound, but the paper's new families rest on 'similar' proofs; Theorem 3.9's count is false as stated (N(G,1)=0 not m(s−1) when m∤1), leaving the central constructions unproven as written.","rationale":"The generic construction in Theorem 2.3 is proven directly and appears correct: the collision set for the coset index function is exactly the union over g∈G of the solution sets of x(g−1)=a, and the image size formula m=Σ M(G,a)/a is consistent with counting cosets. The paper's contribution, however, is the concrete families in Section 3, and these are supported by terse 'proof is similar' arguments. In Theorem 3.9 the stated count is false as written: for n=12 it gives N(G,1)=0, not 4, because g−1 is divisible by m and α=1 is not divisible by m. The final S set survives in the tested case, but the proof's reasoning is invalid. The same unverified counting appears in Theorems 3.6, 3.7, and 3.10, where the counts N(G,α) are asserted without derivation. This matches the reader's weakest assumption. Because the generic framework seems sound and the theorem statements may be true after correcting the counts, the appropriate disposition is conditional acceptance, unchanged from the reader's verdict.","tokens_in":9272,"tokens_out":26875,"duration_ms":257233,"concrete_test":"Re-derive N(G,α) in Theorem 3.9 via CRT: for n=mp with gcd(m,p)=1, write g−1=m c_g and reduce the congruence modulo p. Show that (g−1)x≡α (mod mp) has zero solutions when m∤α, and has exactly m(s−1) solutions when m|α and p∤α. Then compute N(G,α) exhaustively for all α∈Z_{mp} for parameter sets such as (m,p,s)=(4,3,2), (2,5,2), (3,7,3), and (5,11,5), confirming that every value lies in {0,m(s−1)}. If any outside value occurs, Theorem 3.9 is false; otherwise the theorem survives but the proof needs the stated correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's stated contributions are the concrete ZDF families of Section 3, yet Theorems 3.6, 3.7, 3.9, and 3.10 are justified only by 'proof is similar with that of Theorem 3.5' (Sections 3.D and 3.E). These omitted counts are load-bearing because Theorem 2.3 defines the ZDF parameter set S exactly as the collection of values N(G,a). For Theorem 3.9, the proof's item (4) asserts N(G,α)=m(s−1) for every α not divisible by p. This is false: when m∤α, every g−1 is divisible by m, so (g−1)x≡α (mod mp) has no solutions. The reader's example n=12, m=4, p=3, s=2, G={1,5} gives N(G,1)=0, not m(s−1)=4. The final S set still comes out {0,4} because only α with m|α and p∤α contribute m(s−1), but the written proof does not establish this. The analogous counts in Theorems 3.6, 3.7, and 3.10 are asserted without derivation, so the claimed new parameters are not verified by the text as it stands.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9609,"tokens_out":19763,"duration_ms":192563,"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":[{"comment":"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.","section":"Theorem 3.9, item (4)"},{"comment":"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.","section":"Theorem 3.7, item (4)"},{"comment":"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.","section":"Sections 3.D and 3.E"}],"minor_comments":[{"comment":"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.","section":"Section 3.D"},{"comment":"The word 'communicative' should be 'commutative'; similarly, the conclusion contains the typo 'Serval' for 'Several'.","section":"Remark 3.1"},{"comment":"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.","section":"Theorem 3.7, item (2)"}],"recommendation":"major_revision","confidential_remarks":"The two incorrect intermediate counts are not simply omissions; they are false statements in the current text. Still, the final parameter sets in the tested examples appear salvageable, and the generic construction is sound, so the paper is worth a careful revision rather than rejection. The revision should replace all 'proof is similar' passages in Theorems 3.6, 3.7, 3.9, and 3.10 with complete counting arguments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe trouble is in Section 3, where the new families live. Theorem 3.5 (n = p^2, G = <p−1>) 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.\n\nThe 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.\n\nGiven 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.\n\nWho 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.","headline":"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.","tokens_in":10124,"tokens_out":2662,"would_cite":true,"duration_ms":27173,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B10","11T22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["zero-difference function","coset index function","cyclotomic classes","residue class ring","constant composition code","difference system of sets","frequency-hopping sequence","balanced functions"],"falsifier":"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.","tokens_in":9088,"feed_emoji":"🧮","tokens_out":7276,"duration_ms":71451,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every subgroup of a finite ring gives a zero-difference function","feed_subtitle":"The technique turns the search for zero-difference functions into a counting problem.","key_machinery":"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 $ax\\equiv b\\pmod n$.","core_discovery":"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 $p$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the partition of the ring into cosets $rG$ of a multiplicative subgroup, which defines the coset index function $f_G$.","marker":"[9]"},{"why":"Gives the standard count of solutions to a linear congruence $ax\\equiv b\\pmod n$, used to compute $N(G,a)$ in all the explicit families.","marker":"[10]"},{"why":"Introduced the zero-difference balanced function concept that this paper generalizes to functions whose difference counts lie in a set $S$.","marker":"[4]"},{"why":"Introduced the generalized zero-difference balanced function with a set $S$, which the paper renames as the zero-difference function it studies.","marker":"[12]"},{"why":"Provides the earlier generic construction of zero-difference balanced functions that Corollary 2.4 extends to rings not required to be commutative.","marker":"[16]"}],"fun_headline_variants":["Zero-difference functions via cyclic subgroups in rings","Subgroups of finite rings yield zero-difference functions","Counting cosets builds zero-difference functions","Cyclotomy construction of zero-difference functions","Ring subgroups always give zero-difference functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-difference functions via cyclic subgroups in rings","Subgroups of finite rings yield zero-difference functions","Counting cosets builds zero-difference functions","Cyclotomy construction of zero-difference functions","Ring subgroups always give zero-difference functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000327,"raw_usage":{"total_tokens":1782,"prompt_tokens":853,"completion_tokens":929,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":859}},"tokens_in":469,"tokens_out":929,"duration_ms":9281,"temperature":1.0,"reasoning_tokens":859,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:40:48.299599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Eilenberg, Automata, languages, and machines","cited_arxiv_id":null,"evidence_quote":"Supplies the partition of the ring into cosets $rG$ of a multiplicative subgroup, which defines the coset index function $f_G$."},{"cited_title":"Ireland and M","cited_arxiv_id":null,"evidence_quote":"Gives the standard count of solutions to a linear congruence $ax\\equiv b\\pmod n$, used to compute $N(G,a)$ in all the explicit families."},{"cited_title":"Highly nonlinear mappings,","cited_arxiv_id":null,"evidence_quote":"Introduced the zero-difference balanced function concept that this paper generalizes to functions whose difference counts lie in a set $S$."},{"cited_title":"On generalized zero-difference balanced functions,","cited_arxiv_id":null,"evidence_quote":"Introduced the generalized zero-difference balanced function with a set $S$, which the paper renames as the zero-difference function it studies."},{"cited_title":"A generic method to construct zero-difference balanced functions,","cited_arxiv_id":null,"evidence_quote":"Provides the earlier generic construction of zero-difference balanced functions that Corollary 2.4 extends to rings not required to be commutative."}],"review_version":1}