The minimum distance of the antiprimitive BCH code with designed distance 3

Haojie Xu; Wei Lu; Xia Wu; Xiwang Cao

The minimum distance of the antiprimitive BCH code with designed distance 3

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For antiprimitive BCH codes with designed distance 3, the minimum distance is 3 exactly when gcd(2h+1, q+1, q^m+1) is not 1; when q and m are both odd it is 4 exactly when gcd(2h+1, q+1) equals 1.

desk verdict A clean, correct full characterization for odd q and m, honestly scoped; deserves refereeing, with a few Magma-backed claims to be documented. read the letter →

arxiv 2505.04315 v1 pith:X7VHCBHH submitted 2025-05-07 cs.IT math.IT

Haojie Xu , Xia Wu , Wei Lu , Xiwang Cao This is my paper · ORCID

classification cs.ITmath.IT MSC 94B1594B05

keywords cycliccodeBCHLCDantiprimitivesubfieldsubminimumdistancedistance-optimalrootsofunity

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

This paper determines the exact minimum distance of the antiprimitive BCH code $C(q,q^m+1,3,h)$, a cyclic error-correcting code of length $q^m+1$ whose parity checks involve consecutive powers $h$ and $h+1$. It proves that $d=3$ happens precisely when $\gcd(2h+1,q+1,q^m+1)\ne 1$, and that when both $q$ and $m$ are odd the minimum distance is completely decided by one gcd condition: $d=4$ if and only if $\gcd(2h+1,q+1)=1$, and $d=3$ otherwise. Exact minimum distances of this kind are known for only a few families of BCH codes, and these codes are LCD codes, so the result has direct relevance to cryptographic applications that need linear codes whose duals intersect them trivially. The characterization also yields two infinite families of distance-optimal codes and several codes with best known parameters.

What carries the argument

The argument runs on roots of unity in $\mathbb{F}_{$q^{{2m}}$}$, specifically the set $U_l$ of $l$-th roots of unity, together with two rational expressions: the determinant $D(x,y)=x^h y^h(y-x)$ and the ratio $E(x,y)=($x^{{2h+1}}$-$y^{{2h+1}}$)/(x-y)$. Lemma 3 gives $D(x,y)^{q^m}=-$x^{{-2h-1}}$$y^{{-2h-1}}$D(x,y)$, which turns an $\mathbb{F}_q$-linear dependence among parity-check columns into the equation $(x/z)^{2h+1}=1$. That identity drives the $d=3$ gcd condition. For $d=4$, a dependence of four columns forces the ratio equality $E(x,z)/E(x,w)=E(y,z)/E(y,w)$; when $q$ and $m$ are odd, the paper shows four suitable points always exist by choosing $x, $x^{{-1}}$, 1, -1$ inside $U_{q+1}$, using $$x^{{q^m}}$=$x^{{-1}}$$ and the inclusion $U_{q+1}\subseteq U_{q^m+1}$ that holds for odd $m$.

What would settle it

Compute the true minimum distance for a small instance in the predicted regime; for example, $C(3,$3^{3}$+1,3,1)$ has $\gcd(3,4)=1$, so Theorem 7 requires $d=4$, and an exhaustive syndrome search that finds any other value would refute the claim. Similarly, any $q,m,h$ with $\gcd(2h+1,q+1,q^m+1)>1$ must have $d=3$ by Theorem 4, so a search finding no weight-3 codeword in such a case would falsify that theorem.

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Extended reading notes

Core claim

The central result is a full gcd characterization of the minimum distance of $\mathcal{C}_{(q,q^m+1,3,h)}$. Theorem 4 states that $d=3$ if and only if $\gcd(2h+1,q+1,q^m+1)\ne 1$, which is equivalent to: $m$ is odd and $\gcd(2h+1,q+1)\ne 1$. Theorem 7 completes the picture when both $q$ and $m$ are odd: there $d=3$ if and only if $\gcd(2h+1,q+1)\ne 1$, and $d=4$ if and only if $\gcd(2h+1,q+1)=1$. The three-way gcd collapses because $\gcd(q+1,q^m+1)=q+1$ for odd $m$. For the remaining parity cases the paper supplies partial sufficient conditions for $d=4$, such as $\gcd(h,q^m+1)\ge 3$ or $\gcd(h+1,q^m+1)\ge 3$ when $q$ is odd and $m$ is even.

Load-bearing premise

The complete $d=4$ classification assumes that both $q$ and $m$ are odd, because only then do the needed four roots of unity lie in the required unit group and can be taken as $1, -1, x, $x^{{-1}}$$; outside that parity regime the paper gives only partial conditions for $d=4$.

Editorial extensions

If this is right

For every prime power $q$ and every $m$, the minimum distance is 3 exactly when $m$ is odd and $\gcd(2h+1,q+1)>1$; in every other case the code has minimum distance at least 4.
When $q$ and $m$ are both odd, no intermediate values occur: the minimum distance is 4 when $\gcd(2h+1,q+1)=1$ and 3 otherwise.
For odd $q$, choosing $h=0$ or $h=(q^m+1)/2$ gives codes with parameters $[q^m+1, q^m-2m, 4]$, and for $q>3$ these codes attain the Sphere Packing bound, so they are distance-optimal.
The theorem settles the first part of Conjecture 23 from the earlier LCD-code investigation: $C(3,3^m+1,3,0)$ has minimum distance 4.
For $q=2$ and $h=1$ with $m\equiv 0 \pmod 4$, the minimum distance is 5 when $m\not\equiv 0 \pmod{16}$, and is 5 or 6 when $m\equiv 0 \pmod{16}$.

Reading between the lines

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

The parity restriction looks structural rather than technical: when $m$ is even, $U_{q+1}$ is not contained in $U_{q^m+1}$, so the four-point ratio construction cannot be transplanted from the $q+1$ circle; a full $d=4$ classification for even $m$ likely needs a different family of four points.
The equality $E(x,z)/E(x,w)=E(y,z)/E(y,w)$ is a cross-ratio statement under the rational map $t\mapsto t^{2h+1}$, so a natural next step is to classify all four-point solutions inside $U_{q^m+1}$; Remark 2 shows the condition is necessary but not sufficient, meaning an additional invariant separates $d=4$ from $d\ge 5$.
The same coefficient-fixing method should extend to larger designed distances: replace the two-row parity-check matrix by $\delta-1$ rows and ask when $\delta$ roots of unity admit $\mathbb{F}_q$ coefficients, turning gcd conditions into systems of equations of the form $(x_i/x_1)^{2h+1}=1$.
The remaining $q=2$, $m\equiv 0 \pmod{16}$ case is a concrete computational target: a finite-field search over the $2^m+1$ roots of unity could determine whether the minimum distance is always 5 and, if so, suggest a general construction of weight-5 codewords.

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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

Quick take: the paper settles the minimum distance of the antiprimitive BCH code C(q,q^m+1,3,h) in the odd-q, odd-m case with a clean iff characterization, and gives a general iff condition for d=3. Solid, honest, mainstream BCH-code work; the main theorems deserve refereeing.

What is actually new: Theorem 4/Corollary 1 — for all q and m, d=3 iff m is odd and gcd(2h+1,q+1)≠1 — and Theorem 7, which for odd q and odd m gives d=4 iff gcd(2h+1,q+1)=1, completing the picture. This genuinely extends the authors' earlier length-(q+1) result [20] to length q^m+1, and the proof is their own: I traced the Cramer-rule step, the determinant manipulation behind (10)–(11), and the gcd argument excluding the nonzero case, and it holds. The stress-test note is right that the odd case is internally sound. The self-citations to [20] and [21] are legitimate — [20] is the m=1 special case being generalized, not padding.

I agree with the reader's conditional verdict and the 7 on soundness. Credit also for candor: the concluding remarks state plainly that the even q or even m d=4 problem is not resolved by these techniques.

Soft spots, in proportion. The d=4 characterization is genuinely incomplete for even q or even m: Proposition 8 gives sufficient conditions only, and Remark 5 shows via Magma that they are not necessary. More importantly, Theorem 13 is stated as a theorem, but the m≡8 mod 16 case rests on a Magma verification with no script or witness supplied; that specific claim has a computer-assisted gap. The same applies to the auxiliary Magma checks in Remarks 5 and 7 and Example 1. None of this touches the main characterization, but the authors should either provide the code and outputs or replace the checks with short arguments. One harmless omission: Proposition 6 does not explicitly say the constructed i,j are nonzero; this follows from d≥4 already being established.

Verdict: this deserves a serious referee. The odd case is a correct, self-contained generalization of known results, the paper is honest about scope, and there is useful byproduct material for the code tables. For anyone working on BCH or cyclic codes, this is worth engaging with. I would accept it for review and ask for the computational claims to be documented before final acceptance.

Referee Report

0 major / 6 minor

Summary. The paper studies the minimum distance d of the antiprimitive BCH code C(q,q^m+1,3,h) over F_q. The main results are Theorem 4, which proves d=3 if and only if gcd(2h+1,q+1,q^m+1) is not 1, and Theorem 7, which fully characterizes d for odd q and odd m: d=3 if and only if gcd(2h+1,q+1) is not 1, and d=4 if and only if gcd(2h+1,q+1)=1. The paper also gives sufficient conditions for d=4 in even cases, determines parameters for special values of h (including the binary case q=2), derives two infinite families of distance-optimal codes, and lists several codes with best known parameters.

Significance. If correct, Theorem 7 settles the minimum-distance question for designed-distance-3 antiprimitive BCH codes in the odd-q, odd-m regime, generalizing earlier work for m=1 by Xu et al. The proof strategy, based on Cramer's rule, trace identities, and the BCH and sphere-packing bounds, is self-contained for the central results and is algebraically sound in the cases I traced. The distance-optimal code families in Theorems 9 and 12 are useful additions to the literature. The main limitation is that some auxiliary claims (Theorem 13 and Remarks 5 and 7) depend on Magma computations for which no scripts or explicit hand-verifiable certificates are provided, which reduces reproducibility but does not affect the central theorems.

minor comments (6)

[Theorem 12, proof, Section IV-B] The proof of Theorem 12 splits into the cases "q > 4" and "q = 2", but for even q the value q=4 is never treated. Since the sphere-packing estimate used for q>4 also holds for q=4, the first case should read "q \ge 4" (or "q > 2") and the q=4 case should be explicitly included.
[Theorem 13 and Remarks 5 and 7] Theorem 13 (the case m ≡ 8 mod 16) and Remarks 5 and 7 state that certain minimum distances or root properties are verified by Magma, but no scripts or explicit algebraic certificates are given. These claims are not independently checkable from the manuscript; please provide the verification scripts or replace the computations with hand-verifiable arguments.
[Theorem 10 proof, Section IV-A] In the first case of the d≥5 proof, the displayed equation after (25) contains a repeated term: "xhyh + xhzh + xhyh + yhzh + yhwh + zhwh" should read "x^h y^h + x^h z^h + x^h w^h + y^h z^h + y^h w^h + z^h w^h".
[Proposition 5 proof, Section III] In the expression for j after Cramer's rule, "lwn+1" should be "l w^{h+1}".
[Introduction and Section III] The notation for m is inconsistent: the Introduction says "Let m be the multiplicative order of q modulo n", but for n=q^m+1 the order is 2m. Throughout Section III the parameter m is instead the exponent in n=q^m+1. This should be clarified to avoid confusion.
[Minor typographical issues] There are several typos, including "Now we proof that" in Theorem 4, "Rising" for "Raising" in the proofs of Theorems 10 and 12, "orders pf" in Theorem 14, and a stray "==" in the gcd formula in Theorem 14. Please proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the distance characterizations are derived from parity-check equations and finite-field identities, not from the target statements.

full rationale

The paper's central results are derived in a self-contained way from the parity-check matrix of the antiprimitive BCH code, Cramer's rule, the Frobenius action on roots of unity, and elementary gcd arguments. In Theorem 4, the d=3 characterization is proved in both directions: d=3 is used to derive a nontrivial common divisor of 2h+1, q+1, and q^m+1 via the equations forced by a three-column dependency, and the converse constructs such a dependency explicitly from three distinct elements of the gcd subgroup. The d=4 result for odd q and m is likewise derived: Proposition 6 proves a sufficient condition using the ratio identity E(x,z)/E(x,w)=E(y,z)/E(y,w), and Theorem 7 supplies the required four distinct elements x, x^{-1}, 1, -1 in U_{q+1} by a direct algebraic calculation. No fitted parameter is renamed as a prediction, and no hypothesis equivalent to the desired minimum distance is smuggled into the proof. The self-citations to [20] are used only for comparison and generalization, for example in Remarks 1 and 4, and the proofs of Theorems 4 and 7 do not rely on those citations as premises. The Magma-dependent statements, such as Remark 5, Theorem 13 for m≡8 mod 16, and Example 1, are explicitly presented as computational checks or conjectures rather than as load-bearing derivations, so they do not constitute circularity. The acknowledged limitations for even q or m are scope restrictions, not hidden assumptions. Overall, the derivation chain is independent of its conclusions, and the paper exhibits no self-definitional, fitted-input, or self-citation-load-bearing circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim is derived from finite-field definitions and standard coding bounds. No free parameters are fitted; q, m and h are inputs of the code family. No new entities are postulated.

assumptions (6)

standard math BCH bound: a BCH code with designed distance delta has minimum distance at least delta.
Invoked throughout to get d ≥ 3 and d ≥ 4 in Sections III and IV.
standard math Sphere-Packing bound (Theorem 2 in the paper).
Used to prove distance-optimality in Theorems 9 and 12.
standard math Theorem 1 from Massey and Yang: a cyclic code is LCD iff its generator polynomial is self-reciprocal, and antiprimitive BCH codes are reversible.
Used in the introduction to motivate the LCD property; not directly needed for the main distance proofs.
standard math Subfield subcode properties and the q-cyclotomic coset size dividing 2m (Theorem 3).
Used to compute dimensions in Theorems 9, 12 and 14.
domain assumption For x in U_{q^m+1}, x^{q^m} equals x^{-1}.
Used throughout Lemmas 3, 4 and the proofs of Theorems 4 and 7; valid because x^{q^m+1}=1.
standard math gcd(q+1,q^m+1) equals q+1 when m is odd, and equals 1 or 2 when m is even.
Used in Corollary 1 and Theorem 7 to simplify the gcd conditions.

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Cite this review

Pith. "Pith review of The minimum distance of the antiprimitive BCH code with designed distance 3." pith.science (2026). https://pith.science/paper/X7VHCBHH

@misc{pith2026250504315,
  author       = {Pith},
  title        = {Pith review of: The minimum distance of the antiprimitive BCH code with designed distance 3},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7VHCBHH}},
  note         = {Machine review of arXiv:2505.04315}
}

abstract

Let $\mathcal{C}_{(q,q^m+1,3,h)}$ denote the antiprimitive BCH code with designed distance 3. In this paper, we demonstrate that the minimum distance $d$ of $\mathcal{C}_{(q,q^m+1,3,h)}$ equals 3 if and only if $\gcd(2h+1,q+1,q^m+1)\ne1$. When both $q$ and $m$ are odd, we determine the sufficient and necessary condition for $d=4$ and fully characterize the minimum distance in this case. Based on these conditions, we investigate the parameters of $\mathcal{C}_{(q,q^m+1,3,h)}$ for certain $h$. Additionally, two infinite families of distance-optimal codes and several linear codes with the best known parameters are presented.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [20]

H. Xu, X. Wu, W. Lu and X. Cao, ”The sufﬁcient and necessar y conditions for the minimum distance of the BCH code C(q,q+1 ,3,h) to be 3 and 4,” IEEE Trans. Inf. Theory. , doi: 10.1109/TIT.2025.3535700

arXiv 2025

[2] [21]

The dual codes of two famili es of BCH codes,

H. Xu, X. Wu, W. Lu and X. Cao, “The dual codes of two famili es of BCH codes,” 2024, https://doi.org/10.48550/arXiv.2 410.15070

doi:10.48550/arxiv.2 2024

[3] [1]

Almost MDS codes,

M.A.D. Boer, “Almost MDS codes,” Des. Codes Cryptogr ., vol. 9, no.2, pp. 143-155, Oct. 1996

work page 1996

[4] [2]

Explicit evaluations of some Weil sums,

R.S. Coulter, “Explicit evaluations of some Weil sums,” Acta Arith. , vol.83, no.3, pp. 241-251, 1998

work page 1998

[5] [3]

On the evaluation of a class of Weil sums in characteristic 2,

R.S. Coulter, “On the evaluation of a class of Weil sums in characteristic 2,” New Zealand J. Math. , vol. 28, no.2, pp. 171-184, 1999

work page 1999

[6] [4]

Complementary dual codes for c ounter-measures to side-channel attacks,

C. Carlet and S. Guilley, “Complementary dual codes for c ounter-measures to side-channel attacks,” in: Coding Theory and Applications , CIM Series in Mathematical Sciences, vol. 3, R. Pinto, P . Rocha Malonek , P . V ettori, Eds. Springer, Cham, 2015

work page 2015

[7] [5]

On subﬁeld subcodes of modiﬁed Reed-Solom on codes (Corresp.),

P . Delsarte, “On subﬁeld subcodes of modiﬁed Reed-Solom on codes (Corresp.),” IEEE Trans. Inf. Theory , vol. 21, no. 5, pp. 575-576, 1975

work page 1975

[8] [6]

On near-MDS codes,

S. Dodunekov and I. Landgev, “On near-MDS codes,” J. Geom. , vol. 54, nos.1-2, pp. 30-43, Nov. 1995

work page 1995