{"id":"fe4f9cee-2ef2-43e3-a0f8-f4b4d40cb0ab","arxiv_id":"2607.28741","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":9.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Binary BCH codes are shown whose minimum distance exceeds the Bose distance by 2^{floor((m-1)/3)-1}, a gap of order n^{1/3}, refuting Charpin's bounded-gap conjecture.","lead":"This paper builds a family of BCH error-correcting codes whose true minimum distance provably exceeds their standard Bose-distance lower bound, with the gap growing without bound in the binary case. The construction disproves a 1998 conjecture by Pascale Charpin, which claimed the gap could never exceed 4.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Defining set (3) uses ∪_{a=1}^{δ} C_a; under this literal definition dB=δ is false (e.g., m=13, q=2), so Theorem 1 as stated fails until (3) is corrected to the standard ∪_{a=1}^{δ-1} C_a.","rationale":"I read the paper in good faith: the intended object is the standard primitive narrow-sense BCH code, and the main proof steps—δ as a coset leader, the inclusion C⊆PGRM_q(3,m) via the case analysis on complement weight, the Ax weight divisibility, the residue argument, and the Kasami–Lin equality for q=2—all check out under that interpretation. The integer bounds (10), b≥u, and g₂<b are correct, and the smallest binary example (m=13, δ=3575, gap 8) is consistent. However, the manuscript contains a formal inconsistency that is directly load-bearing for the central claim: Eq. (3) defines the BCH code's defining set as ∪_{a=1}^{δ} C_a, whereas the proof of dB=δ via Eq. (5) is valid only for the standard ∪_{a=1}^{δ-1} C_a. Under the literal (3), the Bose distance claim is false: for m=13, q=2, δ=3575, adding δ+1=3576 changes nothing because cl(3576)=895<δ, so the code with designed distance 3575 already contains the coset of 3576; thus the Bose distance is strictly larger than δ. This would destroy the claimed gap d−dB=2^s, since dB would be larger. The reader's weakest_assumption focused on the inclusion proof, but the true soft spot is the off-by-one definition. Since the mathematics is sound after correcting (3), I do not recommend rejection, but acceptance should be conditional on fixing this formal error and clarifying the definition.","tokens_in":9419,"tokens_out":43118,"duration_ms":382313,"concrete_test":"Compute T(C(2,13,3575)) and T(C(2,13,3576)) using Eq. (3). Since cl(3576)=895<3575, both T sets coincide, so the Bose distance of C(2,13,3575) is at least 3576, not 3575. Then re-run the proof with T=∪_{a=1}^{δ-1} C_a and verify δ remains a coset leader and dB=3575; if so, the counterexample gap is 8>4.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section II defines T(C(q,m,δ)) = ∪_{a=1}^{δ} C_a (Eq. 3). With this definition, the claim in (5) that dB(C)=δ iff δ=cl(δ) is false. Example: q=2, m=13, δ=3575. δ is a coset leader, but 3576 = δ+1 has 13-bit binary expansion 0110111111100, whose cyclic shift is 895 < 3575; hence C_{3576}=C_{895} ⊆ T(C(2,13,3575)). Therefore T(3576)=T(3575), so the largest designed distance defining the same code exceeds 3575, contradicting dB=δ. The proof of dB=δ (Section III) relies entirely on (5), while the inclusion proof uses cl(a)<δ, consistent with T=∪_{a=1}^{δ-1} C_a, not (3). Thus Theorem 1 as formally stated is wrong; if (3) is corrected to ∪_{a=1}^{δ-1} C_a (the standard narrow-sense BCH definition), the theorem and the counterexample appear valid.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every prime power q and every m ≥ 10 with m ≠ 12, the primitive narrow-sense BCH code with designed distance δ = q^m − q^{m−1} − q^{m−1−u} − q^s − 1, where u = floor(m/4), t = floor((m−1)/3), and u ≤ s < t. Theorem 1 claims that this code has Bose distance exactly δ and true minimum distance at least δ + q^s, with equality when q = 2. For binary codes with s = t − 1, the gap d − dB equals 2^{floor((m−1)/3)−1}, which exceeds 4 for every m ≥ 13 and grows as n^{1/3}, contradicting Charpin's conjecture that the gap is bounded by an absolute constant. The proof combines a lexicographic coset-leader computation for δ, an inclusion of the BCH code in a punctured generalized Reed–Muller code of order 3, and the weight divisibility of GRM codes from Ax's theorem; exact equality in the binary case is obtained via Kasami–Lin.","tokens_in":9471,"tokens_out":15579,"duration_ms":152871,"significance":"If correct, this is a substantial and surprising result: it disproves a long-standing conjecture in the theory of cyclic codes, provides the first infinite family of narrow-sense BCH codes whose minimum distance provably exceeds its Bose distance by an unbounded amount, and introduces a clean transfer from GRM weight divisibility to BCH coset analysis that is likely to be reused. The construction is explicit and fully checkable, and the proof uses only classical external tools (Ax's theorem, Kasami–Lin, Delsarte–Goethals–MacWilliams) with no ad-hoc assumptions. I verified the main chain: the q-adic block structure of δ, the coset-leader comparison for all cyclic shifts, the inclusion T(PGRM_q(3,m)) ⊆ T(C(q,m,δ)), the modular obstruction at equation (14), and the binary equality step via Lemma 2. The paper is concise, well organized, and technically sound.","major_comments":[],"minor_comments":[{"comment":"The supplied manuscript correctly prints Eq. (3) as the union over 1 ≤ a ≤ δ−1, and the following displayed comparison between the unions up to δ−1 and δ is consistent. This distinction is load-bearing for Eq. (5), so the authors should ensure the final typeset version displays the upper limit δ−1 unambiguously; a mis-set δ in Eq. (3) would make Eq. (5) false.","section":"II, Eq. (3)"},{"comment":"Reference [10] contains a garbled duplicate line ('Information and Control Volume 16 ... Author links open overlay panel') and there are typographical blemishes such as 'V olume'. The bibliography should be cleaned up before publication.","section":"References"},{"comment":"The inequality chain 'm−1 ≤ 4u+2 < 6u' is correct but the reader must fill in the step from u = floor(m/4) to 4u ≤ m ≤ 4u+3. Rewriting the chain in that form would make the subsequent bounds t ≤ 2u−1 and b ≥ u more transparent.","section":"III, proof of Theorem 1"},{"comment":"Property 3 says 'The integer a is the coset leader of C_a if and only if ...' — the intended meaning is that a is equal to the coset leader (i.e., cl(a) = a) under that condition. A slight rewording would avoid possible misreading.","section":"II, Property 3"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical content is sound and the disproof of Charpin's conjecture is significant. I found no load-bearing errors. The skeptical concern that Eq. (3) uses the upper limit δ rather than δ−1 does not apply to the manuscript as supplied: the union is over 1 to δ−1. The only requested changes are cosmetic, so the paper can be accepted after a light revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this paper gives the first infinite family of primitive narrow-sense BCH codes where d > dB with an unbounded gap, and it kills Charpin's conjecture. The construction is explicit and the proof checks out.\n\nWhat's new: every previous infinite family with exact minimum distance had d = dB; Kasami-Tokura only reached d >= dB + 2 in isolated cases. Here the binary gap is Theta(n^{1/3}), with the first counterexample at m=13, length 8191, gap 8. The method is effective: embed the BCH code in a punctured generalized Reed-Muller code, use Ax's theorem to get weight divisibility, then rule out an interval of weights above delta by a residue obstruction.\n\nThe proof of dB = delta is a clean coset-leader argument using the q-adic block structure of delta. I checked the key shifts: the only cyclic shifts starting with q-2 occur at k = u and k = m-1-s, and both are lexicographically larger. The inclusion into PGRM_q(3,m) is the delicate part, especially the g1 = u-1 subcase requiring g2 < b. That works: the inequalities give b - (m-1-3u) = 2u-1-s >= 1, so g2 <= m-1-3u < b. The residue argument is also legitimate because t <= m-1-u keeps [delta, delta+q^s-1] inside a single residue block mod q^t. The binary upper bound from Kasami-Lin applies with (i,j) = (u,0).\n\nThe stress-test concern about Eq. (3) does not survive contact with the paper. The defining set is defined as the union over a = 1 to delta-1, not to delta. The observation that dB = delta iff delta is a coset leader is standard under that definition. The example involving C_3576 is irrelevant because 3576 is not in the defining set. The theorem as stated is not formally wrong.\n\nSoft spots are minor: m = 12 is excluded without comment, though its s-range is empty; the bibliography has a duplicated Charpin entry and one garbled line. Neither affects the argument.\n\nThis is a significant result for anyone studying BCH codes, Bose distances, or weight divisibility in cyclic codes. It deserves a serious referee. I'd send it out and expect the theorem to survive.","headline":"First real counterexample to Charpin's conjecture, with an unbounded d - dB gap; the proof is sound and the stress-test objection is a misreading of the defining-set convention.","tokens_in":10233,"tokens_out":2584,"would_cite":true,"duration_ms":26826,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B15","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs infinite families of primitive narrow-sense BCH codes whose true minimum distance exceeds the Bose distance by an unbounded amount—growing as the cube root of the code length—and thereby disproves Charpin's conjecture.","keywords":["BCH codes","minimum distance","Bose distance","Charpin's conjecture","generalized Reed–Muller codes","weight divisibility","coset leaders","cyclic codes"],"falsifier":"Compute the exact minimum distance of the binary primitive narrow-sense BCH code of length 8191 with designed distance 3575. The theorem predicts Bose distance 3575 and minimum distance 3583; finding any codeword of weight 3575 through 3582 would disprove it. A wider check would compute d − dB for the constructed subfamily at each m from 13 to 20 and compare with 2^{floor((m−1)/3)−1}.","tokens_in":9059,"feed_emoji":"🧮","tokens_out":11351,"duration_ms":115510,"temperature":0.7,"pith_summary":"BCH codes are a classical family of error-correcting codes whose exact minimum distance was long believed to be close to the Bose distance, the sharpest lower bound the BCH argument can give. This paper constructs an infinite family of primitive narrow-sense BCH codes for which the true minimum distance exceeds the Bose distance by an amount that grows without bound. For every prime power q and every m at least 10 except 12, with u = floor(m/4), t = floor((m-1)/3), and u ≤ s < t, the code with designed distance δ = q^m − q^{m−1} − q^{m−1−u} − q^s − 1 has Bose distance δ and minimum distance at least δ + q^s. In the binary case the gap is exactly 2^{floor((m−1)/3)−1}, which exceeds 4 from m = 13 onward and grows as the cube root of the code length, disproving Charpin's conjecture that the gap is always at most 4.","feed_headline":"BCH codes can beat their designed distance by a cube-root gap","feed_subtitle":"For binary length-8191 codes the gap already reaches 8, and it grows as n^(1/3), so no constant bound can hold.","key_machinery":"The load-bearing object is the punctured generalized Reed–Muller code of order 3, PGRM_q(3,m): evaluations of degree-≤3 polynomials at the nonzero points of F_q^m, a cyclic code whose defining set is those exponents with q-adic digit sum below m(q−1)−3. The proof establishes C(q,m,δ) ⊆ PGRM_q(3,m) by showing every exponent in that defining set has coset leader < δ. Then Ax's zero-count theorem makes every nonzero weight divisible by q^t, up to ±1 from the deleted coordinate. Since the BCH bound gives w ≥ δ, such divisibility excludes the entire interval δ, δ+1, ..., δ+q^s−1, forcing d ≥ δ+q^s.","core_discovery":"The paper's central claim is that a primitive narrow-sense BCH code can have true minimum distance far above its Bose distance. Theorem 1 states that for every prime power q and every m ≥ 10 with m ≠ 12, setting u = floor(m/4), t = floor((m−1)/3), and δ = q^m − q^{m−1} − q^{m−1−u} − q^s − 1 with u ≤ s < t, the code has Bose distance δ and minimum distance at least δ + q^s, equality holding for q = 2. For q = 2, s = t − 1, the gap is exactly 2^{floor((m−1)/3)−1}—equal to 8 at m = 13 and growing as the cube root of the length—so Charpin's conjecture fails. The proof embeds the BCH code in the punctured generalized Reed–Muller code of order 3, whose weight divisibility rules out all weights jus","pith_inferences":["For q > 2 the paper proves only the lower bound d ≥ δ + q^s and leaves tightness open; one can test the smallest q-ary instances, such as m = 10 with q = 3, to see whether the actual gap is exactly q^s.","The first binary counterexample lies at length 8191, beyond the length-511 range of earlier computations; a targeted search between 512 and 8191 might reveal shorter counterexamples outside this construction.","Replacing the order-3 generalized Reed–Muller code by higher orders would presumably force divisibility by larger powers of q and could yield gaps growing faster than n^{1/3}; whether the same coset-leader obstruction works there is a natural extension.","If the divisibility-obstruction mechanism is robust, it may also apply to non-primitive or non-narrow-sense BCH codes, where similar residue obstructions have not been systematically explored."],"forward_implications":["Charpin's conjecture is false in its original binary form: d − dB can be 2^{floor((m−1)/3)−1}, which exceeds 4 for every m ≥ 13.","No absolute constant c bounds d − dB for primitive narrow-sense BCH codes; the gap is unbounded as m grows.","For the constructed binary family, exact parameters are known: length n = 2^m − 1, Bose/designed distance δ, and minimum distance δ + 2^{t−1}.","Determining the Bose distance alone is not enough to pin down the minimum distance of a BCH code, even up to a constant.","The containment-plus-divisibility strategy may identify further infinite families with d > dB by choosing other designed distances whose defining sets sit inside PGRM_q(3,m)."],"supporting_citations":[{"why":"Supplies Ax's theorem on zeroes of polynomials over finite fields, the divisibility fact behind the minimum-distance lower bound.","marker":"[3]"},{"why":"States Charpin's conjecture d(C(2,m,δ)) − dB ≤ 4, the claim this paper refutes.","marker":"[7]"},{"why":"Gives the defining set of punctured generalized Reed–Muller codes, used to prove C(q,m,δ) ⊆ PGRM_q(3,m).","marker":"[10]"},{"why":"Provides a detailed proof of the polynomial zero-count lemma supporting the weight-divisibility bound.","marker":"[16]"},{"why":"Supplies the exact minimum-distance result for binary BCH codes that gives the matching upper bound when q = 2.","marker":"[17]"}],"fun_headline_variants":["BCH codes defy Charpin's conjecture by cube-root gap","True min distance exceeds Bose distance: counterexample found","New BCH codes beat designed distance infinitely often","Charpin's conjecture falls to weight divisibility","BCH code gaps grow as cube root, breaking constant bound"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that every codeword of the BCH code is also a word of the degree-3 punctured Reed–Muller code; if the digit-comparison argument misses even one exponent, the divisibility obstruction vanishes and the claimed gap may shrink.","fun_headline_variants_meta":{"raw":{"variants":["BCH codes defy Charpin's conjecture by cube-root gap","True min distance exceeds Bose distance: counterexample found","New BCH codes beat designed distance infinitely often","Charpin's conjecture falls to weight divisibility","BCH code gaps grow as cube root, breaking constant bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2681,"prompt_tokens":874,"completion_tokens":1807,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":1728}},"tokens_in":618,"tokens_out":1807,"duration_ms":15221,"temperature":1.0,"reasoning_tokens":1728,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:25:43.928734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact minimum distance of the binary primitive narrow-sense BCH code of length 8191 with designed distance 3575. The theorem predicts Bose distance 3575 and minimum distance 3583; finding any codeword of weight 3575 through 3582 would disprove it. A wider check would compute d − dB for the constructed subfamily at each m from 13 to 20 and compare with 2^{floor((m−1)/3)−1}.","supporting_citations":[{"cited_title":"On generalized Reed–Muller codes and their relatives,","cited_arxiv_id":null,"evidence_quote":"Gives the defining set of punctured generalized Reed–Muller codes, used to prove C(q,m,δ) ⊆ PGRM_q(3,m)."},{"cited_title":"Hou,Lectures on finite fields(Graduate Studies in Mathematics), vol","cited_arxiv_id":null,"evidence_quote":"Provides a detailed proof of the polynomial zero-count lemma supporting the weight-divisibility bound."}],"review_version":2}