{"id":"0c152f0c-cc54-425b-b9cb-5e366032db35","arxiv_id":"1908.03070","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New infinite families of optimal p-ary cyclic codes with parameters [2(p^m-1)/(p-1), 2(p^m-1)/(p-1)-2m, 4] are constructed for every odd prime p.","lead":"New families of error-correcting cyclic codes over prime fields are constructed with the best possible minimum distance for their length and size. The paper extends a line of optimal-code constructions from ternary alphabets to all odd primes, using a number-theoretic criterion for choosing the code roots.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 cannot hold for p>3: v=p^k+1 gives v≡2 mod (p−1)/2, so Lemma 2.4's condition 2 fails before the proof starts.","rationale":"Good-faith reading: the paper's main machinery is Lemma 2.4 and Theorem 3.4, and the algebraic reductions in Theorem 3.4 appear sound aside from minor omitted nonzero justifications. However, the first construction Theorem 3.1 contains a serious congruence error: since p≡1 mod (p−1)/2, the exponent p^k+1 is always congruent to 2 modulo (p−1)/2, never to 1 unless p=3. The proof's verification of Lemma 2.4 condition 2 is therefore false. Example 3.2 illustrates the failure, and the direct solution at x=−1 shows the −1 exemption in Lemma 2.4 cannot be invoked. This does not disprove Theorem 3.4 or the three corollaries, but it invalidates one of the two advertised families and the p=5 example. The reader's conditional verdict should be retained, but with increased severity: Theorem 3.1 must be corrected, restricted to p=3, or removed, and the examples regenerated.","tokens_in":7199,"tokens_out":49939,"duration_ms":493438,"concrete_test":"Run the advertised example p=5,m=3,k=2 through Lemma 2.4: compute v=p^k+1=26 and evaluate v mod (p−1)/2=2; it is 0, not 1. Then directly test the Lemma 2.3 equation at x=−1, α=3: 2^26−((−1)^26+3)=4−4=0 in F_5, so the −1 exemption used by Lemma 2.4 is invalid. This single check refutes the proof of Theorem 3.1 for p>3; if the authors intended a different exponent, the new definition of v must be stated and the congruence v≡1 mod (p−1)/2 rechecked.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3.1's proof tries to verify condition 2 of Lemma 2.4 with the line \"v≡k+1≡1 (mod (p−1)/2)\". But v=p^k+1 and p≡1 mod (p−1)/2, so v≡2 mod (p−1)/2. For p>3, (p−1)/2>1, so the congruence v≡1 mod (p−1)/2 is impossible; condition 2 of Lemma 2.4 is not satisfied. In Example 3.2 (p=5,m=3,k=2), the stated formula gives v=5^2+1=26 and the printed v=10; both are 0 mod 2, not 1. Moreover, x=−1, α=3 directly satisfies the Lemma 2.3 equation in F_5, since 2^26−((−1)^26+3)=4−4=0, so the Lemma 2.4 exemption of −1 is unavailable. Thus the first family is unproved for every p>3, and the theorem's hypothesis is inconsistent with the criterion it invokes. Only p=3, where (p−1)/2=1, escapes this objection. The Theorem 3.4 construction and its corollaries appear to be unaffected, but the paper's advertised first class of p-ary codes and Example 3.2 are not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes constructions of optimal p-ary cyclic codes with two zeros, of parameters [n, n-2m, 4] where n=2(p^m-1)/(p-1). The authors use a sufficient condition from Liao et al. (Lemmas 2.3 and 2.4) and verify it for two families of exponents v: Theorem 3.1 takes v=p^k+1 under the hypotheses gcd(m,k)=gcd(m,p-1)=1 and (p-1)/2 | k; Theorem 3.4 takes any solution v of (p^t-1)v ≡ p^s-p^h (mod p^m-1) satisfying the stated gcd and congruence conditions, from which three corollaries and several examples are derived.","tokens_in":7412,"tokens_out":13381,"duration_ms":131291,"significance":"Optimal two-zero cyclic codes are a well-studied topic, and a genuinely new family with parameters meeting the sphere-packing bound would be a useful contribution. The proof strategy reduces the no-solution condition in Lemma 2.4 to gcd computations, and the gcd arguments in Theorem 3.4 are essentially reconstructible; Examples 3.6 and 3.8 are consistent with the stated formulas. The significance is weakened by a serious error in Theorem 3.1 and by inconsistent numerical examples, so the contribution currently rests on Theorem 3.4 alone.","major_comments":[{"comment":"Condition 2 of Lemma 2.4 cannot be satisfied for p>3. In the proof, the congruence 'v≡k+1≡1 (mod (p-1)/2)' is arithmetically wrong: because p≡1 (mod (p-1)/2), v=p^k+1≡2 (mod (p-1)/2), independent of k. Since 2≠1 modulo (p-1)/2 whenever p>3, the theorem proves nothing for p>3 and is at best a p=3 statement. Example 3.2, which claims p=5, is therefore outside the stated hypothesis.","section":"Section 3.1, Theorem 3.1 and its proof"},{"comment":"The numerical data in Example 3.2 are inconsistent with the formulas: for p=5, m=3 one has n=62 and v=5^2+1=26, not 10, and the dimension of C_p(1,v) should be n-2m=56, not 50. The displayed generator polynomial has degree 6, consistent with dimension 56, so the stated parameters [62,50,4] need correction.","section":"Example 3.2"},{"comment":"The data in Example 3.10 are also inconsistent: p=3, m=7 gives n=2186 and, for s=3, v=n/2+(3^3-1)/(3-1)=1106; the printed parameters [728,714,4] correspond to n=728, and the symbol 'e=86' is undefined. This example should be recalculated or replaced.","section":"Example 3.10"}],"minor_comments":[{"comment":"The remark asserts that conditions 1) and 2) imply v is even and p≡3 mod 4; this implication is used in the proof of Lemma 2.4 but not proved there. It would be helpful to spell out the argument.","section":"Remark after Lemma 2.3"},{"comment":"There are several typographical errors in the text and references ('monomilas', 'minimun distance', 'Intertional'); these should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The error in Theorem 3.1 is central, not cosmetic. If the authors can restrict the first family to p=3 or replace v with a correctly chosen exponent, and correct the numerical examples, the remaining Theorem 3.4 material may be publishable. The paper would also benefit from stating explicitly that optimality relies entirely on the external criterion from [15]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper quickly: the real content is Theorem 3.4 and its three corollaries. Those appear to be new infinite families of distance-optimal p-ary cyclic codes, and the proofs are reconstructible: the Frobenius manipulation reduces the no-solution condition to gcd statements, and the gcd arithmetic checks out for the cases used. The p=3 special case of Corollary 3.5 generalizes the Ding–Helleseth result, which is a legitimate extension. That part deserves referee time.\n\nThe first class, Theorem 3.1, does not hold as stated. The proof verifies condition 2 of Lemma 2.4 with 'v≡k+1≡1 (mod (p−1)/2)'. But v=p^k+1 and p≡1 mod (p−1)/2, so v≡2 mod (p−1)/2, not 1, for any p>3. The stress-test is right: the hypothesis (p−1)/2 | k makes things worse, because v≡2 is never 1 when (p−1)/2>1. In fact Lemma 2.4's exemption of −1 fails: for p=5, m=3, k=2, x=−1, α=3 solves (x+α)^v − (x^v+α)=0 in F_5. So Example 3.2's claimed optimality is unsupported. This is not a minor typo; it is the paper's advertised 'another class'.\n\nExample 3.10 is also broken: with p=3, m=7, n=2186, the claimed parameters [728,714,4] cannot be right, and 'e=86' matches neither the corollary nor n/2+(p^s−1)/(p−1)=1106. The reader's take flagged this as a conditional issue; it is worse than a typo because it suggests the examples were not computed from the displayed formulas.\n\nThe citation pattern is honest: Lemma 2.4 comes from [15] and is not re-derived, but that is acceptable for a published criterion. The three corollaries do rest on it, so a referee should verify Lemma 2.4 independently. I did not find circular reasoning.\n\nBottom line: this is a mixed paper. The three congruence families are probably a real, useful contribution to the p-ary cyclic-code literature; the first family should be withdrawn or corrected, and the examples need recomputation. For that reason I would send it to a serious referee rather than desk reject, but acceptance hinges on the authors fixing Theorem 3.1 and the examples. As submitted, I would not cite it, and I would not teach the first class.","headline":"Three of the four families look sound, but the advertised v=p^k+1 class is unproved: its congruence check fails for every p>3.","tokens_in":8016,"tokens_out":6961,"would_cite":false,"duration_ms":67843,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B15","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs four families of optimal p-ary cyclic codes with two zeros, all attaining the sphere-packing bound with parameters $[n,n-2m,4]$.","keywords":["cyclic codes","optimal codes","p-ary codes","sphere-packing bound","two zeros","cyclotomic cosets","minimum distance 4","finite fields"],"falsifier":"For $p=5$, $m=5$, $k=2$, Theorem 3.1 predicts an optimal $[1562,1552,4]$ cyclic code with $v=26$; computing the true minimum distance and finding a nonzero codeword of weight below 4 would disprove the theorem. A more direct check is to search $\\Pi\\setminus\\{-1\\}$ and $\\alpha\\in\\mathbb{F}_5^*$ for a solution to $(x+\\alpha)^{26}\\pm(x^{26}+\\alpha)=0$, which the proof asserts cannot exist.","tokens_in":6921,"feed_emoji":"📡","tokens_out":14719,"duration_ms":124589,"temperature":0.7,"pith_summary":"The paper sets out to construct new optimal cyclic error-correcting codes over prime fields, using the 'two zeros' format where the generator polynomial is a product of two minimal polynomials. It claims that for each admissible choice of a prime $p$ and integer $m>2$, with $n=2(p^m-1)/(p-1)$, there are cyclic codes over $\\mathbb{F}_p$ with parameters $[n,n-2m,4]$ that attain the sphere-packing bound and are therefore optimal. The proof runs through two theorems: one for exponents of the form $v=p^k+1$, and one for exponents $v$ solving the congruence $(p^t-1)v\\equiv p^s-p^h\\pmod{p^m-1}$, from which three explicit families follow. A reader should care because optimal cyclic codes combine the strongest possible error correction for their size with the efficient algebraic decoding that cyclic structure provides.","feed_headline":"New optimal p-ary cyclic codes hit the sphere-packing bound","feed_subtitle":"Each code has minimum distance 4 and keeps the efficient encoding and decoding of cyclic codes.","key_machinery":"The central object is the cyclic code $C_p(1,v)$, whose generator polynomial is the product $m_1(x)m_v(x)$ of two minimal polynomials; this is what 'two zeros' means. The device that carries the proof is Lemma 2.4, a sufficient criterion taken from [15]: $C_p(1,v)$ is an optimal $[n,n-2m,4]$ code if $v\\notin C_1$, the cyclotomic coset of $v$ has length $m$, $\\gcd(v-1,n)=1$, $v\\equiv1\\pmod{(p-1)/2}$, and the equations $(x+\\alpha)^v\\pm(x^v+\\alpha)=0$ have no solution $x\\in\\Pi\\setminus\\{-1\\}$ for any $\\alpha\\in\\mathbb{F}_p^*$. The new work is to verify that no-solution condition for specific $v$: Theorem 3.1 does it for $v=p^k+1$ by raising a hypothetical solution to the $p^k-1$ power, and Theorem 3.4 does it for solutions of the congruence above by raising to the $p^t-1$ power and reducing the obstruction to $x_0^{p^s-p^h}=1$ or $x_0^{p^h-v}=1$, each excluded by a gcd condition.","core_discovery":"The central claim is that the cyclic code $C_p(1,v)$ of length $n=2(p^m-1)/(p-1)$ over $\\mathbb{F}_p$ is an optimal $[n,n-2m,4]$ code whenever $v$ satisfies the conditions of Lemma 2.4: $v\\notin C_1$, the cyclotomic coset of $v$ has length $m$, $\\gcd(v-1,n)=1$, $v\\equiv1\\pmod{(p-1)/2}$, and the equations $(x+\\alpha)^v\\pm(x^v+\\alpha)=0$ have no solution in $\\Pi\\setminus\\{-1\\}$ for any $\\alpha\\in\\mathbb{F}_p^*$. The paper supplies new exponents $v$ that pass this test. Theorem 3.1 shows it for $v=p^k+1$ under $\\gcd(m,k)=\\gcd(m,p-1)=1$ and $(p-1)/2\\mid k$ with $m>2$ odd. Theorem 3.4 shows it for any solution $v$ of $(p^t-1)v\\equiv p^s-p^h\\pmod{p^m-1}$ with $\\gcd(m,t)=\\gcd(m,s-h)=1$, $\\gcd(m,p-1)\\mid2$, plus the additional coprimality and congruence conditions. Corollaries 3.5, 3.7, and 3.9 turn these into explicit families, and the examples give concrete codes, including a $[62,50,4]$ code over $\\mathbb{F}_5$, a $[5602,5592,4]$ code over $\\mathbb{F}_7$, and a $[728,714,4]$ code over $\\mathbb{F}_3$.","pith_inferences":["The same exponent-raising method could be tried on other congruences of the form $(p^t-1)v\\equiv p^s-p^h$ with shifted exponents or coefficients; the paper does not explore those, and each candidate would still have to pass Lemma 2.4's no-solution check.","The paper does not compute the weight enumerators of the new codes or their duals; for earlier two-zero cyclic codes such enumerators have often been tractable, so the new families are natural candidates for that analysis.","A finite-field search over small primes and exponents could test whether the gcd conditions in Theorems 3.1 and 3.4 are also necessary for the no-solution condition to hold, or merely sufficient."],"forward_implications":["Every admissible pair $(p,m)$ yields a cyclic code with parameters $[2(p^m-1)/(p-1), 2(p^m-1)/(p-1)-2m, 4]$ over $\\mathbb{F}_p$ that meets the sphere-packing bound.","Because the codes are cyclic, they keep the efficient algebraic encoding and decoding typical of cyclic codes; minimum distance 4 means every single symbol error is correctable and up to three errors are detectable.","The $p=3$ case of Corollary 3.5 recovers a previously known ternary optimal cyclic code family as a special case and extends it to every odd $m\\ge3$ satisfying the stated gcd conditions.","The explicit examples supply generator polynomials for optimal codes over $\\mathbb{F}_5$, $\\mathbb{F}_7$, and $\\mathbb{F}_3$, including a $[62,50,4]$ code over $\\mathbb{F}_5$."],"supporting_citations":[{"why":"Supplies Lemma 2.1 on cyclotomic coset lengths and the Lemma 2.3/2.4 sufficiency criterion that turns the three verifiable conditions on $v$ into optimality of $C_p(1,v)$.","marker":"[15]"},{"why":"Supplies the sphere-packing bound against which the parameters $[n,n-2m,4]$ are judged optimal.","marker":"[11]"},{"why":"Provides the ternary optimal cyclic code result that Corollary 3.5 is said to generalize in the $p=3$ case.","marker":"[4]"}],"fun_headline_variants":["Optimal p-ary cyclic codes from two-zero construction","New families of optimal cyclic codes with distance 4","Two-zero cyclic codes attain the sphere-packing bound","Explicit optimal cyclic codes over finite fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on Lemma 2.4, taken from [15], which asserts that its listed conditions on the exponent $v$ are enough to force the code to have distance 4 and to meet the sphere-packing bound; the paper does not re-prove that lemma, so a hidden flaw or missing hypothesis there would invalidate the optimality claims for every new family.","fun_headline_variants_meta":{"raw":{"variants":["Optimal p-ary cyclic codes from two-zero construction","New families of optimal cyclic codes with distance 4","Two-zero cyclic codes attain the sphere-packing bound","Explicit optimal cyclic codes over finite fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000197,"raw_usage":{"total_tokens":1370,"prompt_tokens":954,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":354}},"tokens_in":570,"tokens_out":416,"duration_ms":4531,"temperature":1.0,"reasoning_tokens":354,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:28:25.544123+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $p=5$, $m=5$, $k=2$, Theorem 3.1 predicts an optimal $[1562,1552,4]$ cyclic code with $v=26$; computing the true minimum distance and finding a nonzero codeword of weight below 4 would disprove the theorem. A more direct check is to search $\\Pi\\setminus\\{-1\\}$ and $\\alpha\\in\\mathbb{F}_5^*$ for a solution to $(x+\\alpha)^{26}\\pm(x^{26}+\\alpha)=0$, which the proof asserts cannot exist.","supporting_citations":[{"cited_title":"Huffman, V","cited_arxiv_id":null,"evidence_quote":"Supplies the sphere-packing bound against which the parameters $[n,n-2m,4]$ are judged optimal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the ternary optimal cyclic code result that Corollary 3.5 is said to generalize in the $p=3$ case."}],"review_version":1}