Counterexamples to Charpin's Conjecture on BCH codes

Maosheng Xiong; Run Zheng; Yaoran Yang; Yutong Zhang

Counterexamples to Charpin's Conjecture on BCH codes

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.28741 v2 pith:KHXGQ5P2 submitted 2026-07-30 cs.IT math.IT

Run Zheng , Yaoran Yang , Yutong Zhang , Maosheng Xiong This is my paper · ORCID

classification cs.ITmath.IT MSC 94B1511T71

keywords BCHcodesminimumdistanceBoseCharpin'sconjecturegeneralizedReed–Mullerweightdivisibilitycosetleaderscyclic

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

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.

What carries the argument

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.

What would settle it

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}.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

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

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.

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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

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.

What'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.

The 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).

The 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.

Soft 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.

This 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.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)

[II, Eq. (3)] 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.
[References] 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.
[III, proof of Theorem 1] 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.
[II, Property 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof relies on external classical theorems and explicit coset-leader arithmetic; the alleged Eq. (3) counterexample misreads the defining set as starting at δ instead of δ−1.

full rationale

The derivation chain is self-contained. The Bose-distance claim dB(C(q,m,δ))=δ is proven by an explicit q-adic digit comparison showing δ is a coset leader (Section III), using only Property 3 and the standard equivalence (5); no fitted parameter is involved. The central lower bound d(C(q,m,δ))≥δ+q^s follows from the inclusion C(q,m,δ)⊆PGRM_q(3,m), which is established by a direct case analysis on q-adic digits, and from the weight divisibility Corollary 1, which is derived from Ax's theorem (Lemma 1), an external classical result. The residue obstruction (14) is a consequence of that divisibility, not an input. The binary equality uses Lemma 2 from Kasami-Lin (1972), also external, together with the set-theoretic inclusion of defining sets. The only self-citation in the paper ([27]) appears in a background list of prior Bose-distance work and is not load-bearing in any proof. The alleged counterexample based on Eq. (3) is not present in the manuscript text supplied: Eq. (3) is ∪_{a=1}^{δ−1} C_a, not ∪_{a=1}^{δ} C_a, so the claimed contradiction does not arise. No step reduces by construction to its own conclusion, and no fitted quantity is renamed as a prediction. Therefore the paper exhibits no significant circularity.

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

The central claim rests on classical theorems, all external to the authors: Ax's zero-count theorem, the BCH bound, the Delsarte-Goethals-MacWilliams defining-set characterization of punctured generalized Reed-Muller codes, and Kasami-Lin's exact minimum-distance result. The constructed parameter δ is an explicit function of q, m, s, not a fitted constant, and no new objects are postulated. One self-citation ([27], by two co-authors) is motivational and does not carry the proof.

assumptions (5)

standard math Ax's theorem (Lemma 1): for a degree-d polynomial over F_q^m, the number of zeros is divisible by q^{ceil(m/d)-1}.
External classical result (Ax 1964, ref [3], detailed proof in Hou [16]); it powers Corollary 1, the weight-divisibility engine for the lower bound.
standard math BCH bound: for designed distance δ, d(C(q,m,δ)) ≥ δ, and the Bose distance is the largest designed distance, still a valid BCH lower bound.
Classical bound invoked in Section III to assert w ≥ δ for any nonzero codeword.
standard math Defining-set characterization of punctured generalized Reed-Muller codes: T(PGRM_q(ℓ,m)) = {a : wt_q(a) < m(q-1) - ℓ} (Eq. 6).
Attributed to Delsarte-Goethals-MacWilliams [10] and Ding-Li-Xia [14]; used to prove C(q,m,δ) ⊆ PGRM_q(3,m).
standard math Kasami-Lin theorem (Lemma 2): for 1 ≤ i ≤ m-j-2 and 0 ≤ j ≤ m-2i, the binary code C(2,m,δ) with δ = 2^{m-1-j} - 2^{m-1-j-i} - 1 has Bose distance and minimum distance equal to δ.
External 1972 result, used only for the binary equality d = δ + 2^s with (i,j) = (u,0); the disproof of the conjecture itself needs only the lower bound.
standard math Coset-leader characterization: dB(C(q,m,δ)) = δ iff δ is a coset leader (Eq. 5), and Property 3: a is a coset leader iff [a]_q is lexicographically minimal among its cyclic shifts.
Standard facts about defining sets of cyclic codes, used to establish dB = δ.

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Pith. "Pith review of Counterexamples to Charpin's Conjecture on BCH codes." pith.science (2026). https://pith.science/paper/KHXGQ5P2

@misc{pith2026260728741,
  author       = {Pith},
  title        = {Pith review of: Counterexamples to Charpin's Conjecture on BCH codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KHXGQ5P2}},
  note         = {Machine review of arXiv:2607.28741}
}

abstract

Determining the exact minimum distance of BCH codes is a longstanding and challenging problem. In this paper, we construct an infinite family of primitive narrow-sense BCH codes whose minimum distance strictly exceeds their Bose distance. Let $q$ be a prime power, let $m$ be an integer with $m \geq 10$ and $m \neq 12$, and set $u = \lfloor m/4 \rfloor$ and $t = \lfloor (m-1)/3 \rfloor$. For each integer $s$ with $u \leq s < t$, we define$$\delta = q^m - q^{m-1} - q^{m-1-u} - q^s - 1.$$We prove that the primitive narrow-sense BCH code with designed distance $\delta$ has Bose distance $\delta$ and a minimum distance of at least $\delta + q^s$, with equality holding for $q = 2$. Furthermore, by setting $s = t - 1$, we derive a subfamily of binary BCH codes in which the gap between the minimum distance and the Bose distance grows at least as the cube root of the code length, strictly exceeding $4$ for all $m \geq 13$. This disproves Charpin's conjecture. We identify these BCH codes by exploiting the weight divisibility properties of generalized Reed--Muller codes.

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Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Studying the locator polynomials of minimum weight codewords of BCH codes,

D. Augot, P. Charpin, and N. Sendrier, “Studying the locator polynomials of minimum weight codewords of BCH codes,”IEEE Trans. Inf. Theory, vol. 38, no. 3, pp. 960–973, May 1992

work page 1992

[2] [2]

Idempotents and the BCH bound,

D. Augot and N. Sendrier, “Idempotents and the BCH bound,”IEEE Trans. Inf. Theory, vol. 40, no. 1, pp. 204–207, Jan. 1994

work page 1994

[3] [3]

Zeroes of polynomials over finite fields,

J. Ax, “Zeroes of polynomials over finite fields,”Amer. J. Math., vol. 86, no. 2, pp. 255–261, 1964

work page 1964

[4] [4]

The weight enumerators for certain subcodes of the second order binary Reed–Muller codes,

E. R. Berlekamp, “The weight enumerators for certain subcodes of the second order binary Reed–Muller codes,”Inf. Control, vol. 17, no. 5, pp. 485–500, Dec. 1970

work page 1970

[5] [5]

On a class of error correcting binary group codes,

R. C. Bose and D. K. Ray-Chaudhuri, “On a class of error correcting binary group codes,”Inf. Control, vol. 3, no. 1, pp. 68–79, Mar. 1960

1960

[6] [6]

A new algorithm for finding minimum-weight words in a linear code: Application to McEliece’s cryptosystem and to narrow-sense BCH codes of length 511,

A. Canteaut and F. Chabaud, “A new algorithm for finding minimum-weight words in a linear code: Application to McEliece’s cryptosystem and to narrow-sense BCH codes of length 511,”IEEE Trans. Inf. Theory, vol. 44, no. 1, pp. 367–378, Jan. 1998. P. Charpin, “Open problems on cyclic codes,” in Handbook Coding Theory, vol. 1, V . Pless and W. C. Huffman, Eds...

work page 1998

[7] [7]

Open problems on cyclic codes,

P. Charpin, “Open problems on cyclic codes,” inHandbook Coding Theory, vol. 1, V . Pless and W. C. Huffman, Eds. Amsterdam, The Netherlands: Elsevier, 1998, ch. 11, pp. 963–1063. 7

work page 1998

[8] [8]

On the minimum distances of some families of BCH codes,

Y . Chen, H. Chen, C. Ding, and H. Lao, “On the minimum distances of some families of BCH codes,”IEEE Trans. Inf. Theory, vol. 72, no. 8, pp. 5736 - 5744, Aug. 2026

work page 2026