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Divisibility of Griesmer Codes

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that for Griesmer codes over $q=p^f$ fields, $q^e$ dividing the minimum distance forces $p^e$ to divide every codeword weight, and adds a geometric divisor theorem that reduces a 2001 conjecture to the small-dimension…

desk verdict A genuine extension of Ward's divisibility theorem for Griesmer codes to non-prime fields, with a few fixable gaps in the write-up. read the letter →

arxiv 2506.07846 v1 pith:V62KRV2E submitted 2025-06-09 math.CO

Haihua Deng , Hexiang Huang , Qing Xiang This is my paper
classification math.CO MSC 11T7111A0794B0594B65
keywords Griesmercodedivisibleboundp-adicvaluationweightpolarizationgeometricapproachconstantsubTeichmüllerlift
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

Griesmer codes are the linear codes whose length attains the theoretical minimum $g_q(k,d)=\sum_{i=0}^{k-1}\lceil d/q^i\rceil$ for given dimension $k$ and minimum distance $d$. This paper proves that when the alphabet has size $q=p^f$ and $q^e$ divides $d$, every codeword weight is divisible by $p^e$. That extends a divisibility law previously known only over the prime field $\mathbb{F}_p$ (for $f=1$) and improves partial results for prime-power alphabets. The proof supplies a structured basis of the code, then uses a $p$-adic valuation criterion; a parallel geometric argument yields the additional divisor $\Delta=\lceil p^{e-(f-1)(q-2)}\rceil$ whenever $p^e$ divides $d$. Together these results prove a weakened form of a 2001 conjecture about divisible Griesmer codes and reduce the full conjecture to the case of dimension at most $q-1$.

What carries the argument

The load-bearing structure is the basis produced by Theorem 4.6: the first $\min\{e+1,k\}$ basis vectors generate a Griesmer subcode in which every nonzero word has weight $d$, and every $k-1$ basis vectors generate a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. On the algebraic side, the proof uses the divisibility criterion of Theorem 5.1, which decides whether $p^e$ divides all weights by checking the $p$-adic valuations of sums of Schur products of Teichmüller lifts; a Teichmüller lift is the unique $(q-1)$-st root of unity congruent to a field element modulo $p$. The critical identity is Proposition 5.9: for $0\le j\le f-1$ and $0\le r\le q-1-p^j$, the $p$-adic valuation of the coefficient $c(r+p^j,p^j)$ equals $\nu_p\binom{r+p^j}{p^j}$. This equality gives exactly the cancellation needed in the backwards induction on the exponents $r_i$. On the geometric side, the proof of Theorem 1.8 is carried by a lemma (Lemma 6.2) that identifies the point multiset restricted to a hyperplane with the residual Griesmer code, so that divisibility of the ambient code descends to hyperplane intersections.

What would settle it

Check Proposition 5.9 by direct computation: for small non-prime $q$ such as $4$, $8$, or $9$, evaluate $\nu_p(c(r+p^j,p^j))$ and $\nu_p\binom{r+p^j}{p^j}$ for every allowed $r$ and $j$, and look for any pair where the valuations differ. As a separate check on the surrounding conjecture, search for a $[g_q(k,d),k,d]_q$ Griesmer code with non-prime $q\ge 8$, $4\le k\le q-1$, and $f+1\le \nu_p(d)<\min\{f(q-2),f(k-1)\}$; finding such a code would disprove the full conjecture that the paper reduces but does not settle.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is Theorem 1.7: if $C$ is a $[g_q(k,d),k,d]_q$ Griesmer code with $q=p^f$ and $q^e\mid d$, then $p^e\mid \mathrm{wt}(c)$ for every $c\in C$; equivalently, if $p^e\mid d$ then $p^{\lfloor e/f\rfloor}$ divides every weight. The authors also prove Theorem 1.8: if $p^e\mid d$, then $\Delta=\left\lceil p^{e-(f-1)(q-2)}\right\rceil$ divides every weight. Both rest on a new basis theorem: a Griesmer code with $q^e\mid d$ has a generator matrix whose first $\min\{e+1,k\}$ rows span a constant-weight Griesmer subcode of minimum weight $d$, and every set of $k-1$ rows spans a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. The algebraic proof of Theorem 1.7 applies a $p$-adic divisibility criterion to Teichmüller lifts of the basis, and the geometric proof of Theorem 1.8 shows that hyperplane intersections inherit divisibility, embedding residual Griesmer codes into hyperplanes of the point multiset.

Load-bearing premise

Everything in the algebraic proof of Theorem 1.7 rests on the carry-counting claim that $\nu_p(c(r+p^j,p^j))$ always equals $\nu_p\binom{r+p^j}{p^j}$ for the stated ranges of $r$ and $j$; if this equality fails for a single pair, the backwards induction that produces $p^e$-divisibility collapses.

Editorial extensions

If this is right

  • If $q^e\mid d$, every code meeting the Griesmer bound over $\mathbb{F}_q$ has all codeword weights divisible by $p^e$; in the equivalent formulation, $p^e\mid d$ forces the divisor $p^{\lfloor e/f\rfloor}$.
  • If $p^e\mid d$, the geometric theorem forces the divisor $\left\lceil p^{e-(f-1)(q-2)}\right\rceil$, which is new whenever this quantity exceeds the divisors known before.
  • The basis theorem gives a structural tool: Griesmer codes with high distance divisibility contain constant-weight Griesmer subcodes generated by the first $e+1$ basis vectors, and all $(k-1)$-generated subcodes are again Griesmer.
  • The full 2001 conjecture is reduced to checking dimension $k\le q-1$, and the paper identifies parameter ranges where a counterexample would most plausibly live.

Reading between the lines

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

  • A natural testable extension is to compute the exact carry structure behind Proposition 5.9 for small non-prime $q$; if the valuation equality holds universally, the same technique might apply to the coefficients $c(r,k)$ needed for the full conjecture.
  • Checking known Griesmer families, such as the unital and ovoid two-weight codes in the examples, against the two divisors would show whether $p^{\lfloor e/f\rfloor}$ or $\lceil p^{e-(f-1)(q-2)}\rceil$ is the tighter bound in practice.
  • The geometric reduction to $k\le q-1$ suggests that finite searches for small non-prime $q$ could settle the conjecture empirically before a general proof is found.
  • The two divisor exponents cross over as $e$ grows: for fixed $f$ and $q$, $\lceil e-(f-1)(q-2)\rceil$ eventually exceeds $\lfloor e/f\rfloor$, so the geometric theorem becomes the stronger statement for large $e$; locating the crossover in known parameters would illuminate where each method dominates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies divisibility properties of Griesmer codes over F_q with q=p^f. Its main results are Theorem 1.7, which states that if q^e divides the minimum distance d, then p^e divides every codeword weight (equivalently, p^{floor(e/f)} divides all weights when p^e divides d), and Theorem 1.8, which states that if p^e divides d, then ceil(p^{e-(f-1)(q-2)}) is a divisor of all codeword weights. The proof strategy is to construct a special basis of the code (Theorem 4.6), apply Ward's weight-polarization criterion to this basis and prove a p-adic valuation statement about certain binomial-coefficient sums (Proposition 5.9), and then use a geometric hyperplane-counting argument for Theorem 1.8. The paper also claims a reduction of Ward's conjecture to the case k <= q-1 (Theorem 6.3) and offers a parameter search for potential counterexamples.

Significance. If the proofs are completed, the paper makes a substantial contribution: it extends Ward's 1998 divisibility theorem for Griesmer codes over prime fields to all prime powers, and it provides a structural basis theorem and a new geometric proof that yields a divisibility result approaching Ward's conjecture. The paper is well organized, uses external results (Ward's criterion and structural propositions) as black boxes rather than circularly, and gives a concrete, falsifiable divisibility statement. The main arithmetic lemma is not machine-checked, however, and the proof of that lemma has a genuine gap as written, so the results cannot currently be regarded as fully established.

major comments (3)
  1. [§5.2, Proposition 5.9, Case 2.2] The proof of the Claim in Case 2.2 invokes Proposition 5.6 with z=(r+p^j)p^{f-j-1-t} and x=b p^f+(p^{f-1-t}-b). Proposition 5.6 requires x,y,z to be nonnegative integers with z=x+y, but for valid parameters, e.g. p=2, f=3, j=1, r=2, b=1, t=0 (so q=8), one has z=8 and x=11, so y is negative and the proposition cannot be applied. In this situation the binomial is zero, so the desired inequality nu_p(·)>s is trivial, but the manuscript does not split off this subcase. Since Proposition 5.9 is used in the Third Step of Theorem 1.7 to divide by the coefficient c(r+p^j,p^j), the proof of Theorem 1.7 is incomplete as written; the missing subcase x>z must be handled explicitly and the remaining digit inequalities checked.
  2. [§5.2, Theorem 1.7, Third Step] The sentence 'by Proposition 5.2 and a similar argument of solving a system of linear equations' conceals a nontrivial step. To isolate the coefficient c(r+p^j,p^j), one must set up the Vandermonde system for the expansion of T(A_1+(α'+β')A_2)^(r+p^j), and unlike in the Second Step, the coefficients c(r+p^j,i) do not all have p-adic valuation zero. The proof must show that after inverting the Vandermonde matrix, no term other than i=p^j can contribute with valuation below s_3. This is exactly the point where Proposition 5.9 is needed, so the argument cannot be left as a 'similar argument'.
  3. [§6, Theorem 6.3] Theorem 6.3 is stated without proof. It is a nontrivial reduction of Ward's conjecture to the case k <= q-1, and Section 7.2 relies on it to formulate Problem 7.1. The phrase 'By the geometric argument in the proof of Theorem 1.8' is not a proof; either the induction for Conjecture 1.6 should be written out, or Theorem 6.3 should be removed or explicitly labeled as a conjecture.
minor comments (3)
  1. [§5.1] There is a typo: 'Ward empolyed' should read 'Ward employed'.
  2. [Theorem 1.8] The ceiling in the statement of Theorem 1.8 is redundant when the exponent is nonnegative; consider stating the divisor as max{1, p^{e-(f-1)(q-2)}} or noting explicitly that the ceiling is only needed when the exponent would be negative.
  3. [§7.2] The suggested search parameters contain a mismatched parenthesis in 'min{f(q−2), f(k−1))}' and the condition 'f+1 ≤ ν_p(d)' should be clarified, as f is already used for the field degree.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the divisibility theorems are proved from Ward's external criterion and structural theorems; the main arithmetic lemma is unverified but not input-equivalent.

full rationale

The paper's central claims (Theorems 1.7 and 1.8) are proved by applying Ward's weight-polarization criterion (Theorem 5.1, from [18]) and Ward's structural results on Griesmer codes (Propositions 4.1 and 4.4, from [19]) to a basis constructed in Theorem 4.6. None of these inputs is defined in terms of the target divisibility: the criterion is a general if-and-only-if characterization of p-power divisibility in terms of additive spanning sets, and the basis theorem is proved from the Griesmer bound and residual-code arguments without assuming the desired weight divisibility. The only new arithmetic content, Proposition 5.9, is a stand-alone lemma asserting equality of p-adic valuations of a coefficient c(r+p^j,p^j) and a binomial coefficient; its proof is a carry analysis that may be delicate, but even if the lemma were false that would be an error in the proof, not a circularity, because the lemma does not encode the conclusion p^e | wt(c) by construction. The proof of Theorem 1.8 uses Theorem 1.7, but Theorem 1.8 is a different statement and the use is a legitimate dependency, not a self-definition. There are no self-citations by the present authors in the reference list; all load-bearing cited results are due to Ward or other named external authors. No fitted parameter is relabeled as a prediction, and no known empirical pattern is merely renamed. Accordingly the derivation chain is self-contained with respect to circularity, and the score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

All inputs are standard prior theorems; no free parameter was fitted and no new entity was postulated. The divisors are pure functions of p, e, f, q, d, and the code dimension. The only non-standard input is Ward's divisibility criterion, which the paper quotes as a black box.

assumptions (7)
  • standard math Griesmer bound (Theorem 1.1): any [n,k,d]_q code satisfies n >= sum_{i=0}^{k-1} ceil(d/q^i).
    Used throughout to identify Griesmer codes and their residual codes.
  • standard math Ward's divisibility criterion (Theorem 5.1 of [18]): p^e is a divisor of C iff the displayed inequality holds for all additive spanning sets.
    The engine of the algebraic proof of Theorem 1.7; quoted as a black box.
  • standard math Ward's Proposition 4.1 (Corollary 5 of [19]): if q^{k-1}|d, then all nonzero codewords of a Griesmer code have weight d.
    Used to identify constant-weight subcodes in Theorem 4.6 and Theorem 1.8.
  • standard math Ward's Proposition 4.4 (Prop 4 of [19]): for any minimum-weight codeword a, there is a supplementary Griesmer subcode.
    The basis construction in Theorem 4.6 depends on this.
  • standard math Kummer's theorem (Theorem 5.3): p-adic valuation of a binomial coefficient equals the number of carries in base p.
    Used in Corollary 5.4, Lemma 5.5, Proposition 5.6, and Proposition 5.9.
  • standard math Teichmuller lift and Schur product machinery from [18], including Proposition 5.2.
    Quoted and used to expand sums in the p-adic proof.
  • domain assumption Linear codes are over F_q with q=p^f and have full length; zero columns are omitted.
    Standard in the geometric multiset setting; zero columns do not affect weights.

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Cite this review

Pith. "Pith review of Divisibility of Griesmer Codes." pith.science (2026). https://pith.science/paper/V62KRV2E

@misc{pith2026250607846,
  author       = {Pith},
  title        = {Pith review of: Divisibility of Griesmer Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V62KRV2E}},
  note         = {Machine review of arXiv:2506.07846}
}
abstract

In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if $p^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. In this paper, we show that if $q^e|d$, then $C$ has a basis consisting of $k$ codewords such that the first $\min\left\{e+1,k\right\}$ of them span a Griesmer subcode with constant weight $d$ and any $k-1$ of them span a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. Using the $p$-adic algebraic method together with this basis, we prove that if $q^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. Based on this fact, using the geometric approach with the aforementioned basis, we show that if $p^e|d$, then $\Delta |{\rm wt}(c)$ for all $c\in C$, where $\Delta=\left\lceil p^{e-(f-1)(q-2)}\right\rceil$.

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Works this paper leans on

22 extracted references · 21 canonical work pages

  1. [1]

    Ax,Zeroes of polynomials over finite fields, Amer

    J. Ax,Zeroes of polynomials over finite fields, Amer. J. Math.86(1964), 255–261

    work page 1964

  2. [2]

    B. I. Belov,A conjecture on the Griesmer boundary, Optimization methods and their applications (All-Union Summer Sem., Khakusy, Lake Baikal, 1972) (Russian), 1974, pp. 100–106, 182

    work page 1972

  3. [3]

    Bonisoli,Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin.18(1984), 181–186

    A. Bonisoli,Every equidistant linear code is a sequence of dual Hamming codes, Ars Combin.18(1984), 181–186

    work page 1984

  4. [4]

    J. H. Conway and N. J. A. Sloane,Sphere packings, lattices and groups, Third, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Math- ematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional con- tributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. 27

    work page 1999

  5. [5]

    S. M. Dodunekov,Optimal linear codes, Mathematics and mathematical education (Bulgarian) (Sunny Beach (Sl ′′nchev Bryag), 1986), 1986, pp. 57–68

    work page 1986

  6. [6]

    S. M. Dodunekov and N. L. Manev,Minimum possible block length of a linear binary code for some distances, Problems Inform. Transmission20(1984), no. 1, 8–14

    work page 1984

  7. [7]

    J. H. Griesmer,A bound for error-correcting codes, IBM J. Res. Develop.4(1960), 532–542

    work page 1960

  8. [8]

    Kawabata and T

    D. Kawabata and T. Maruta,On the nonexistence of ternary linear codes attaining the Griesmer bound, Des. Codes Cryptogr.90(2022), no. 4, 947–956

    work page 2022

Show all 22 references
  1. [9]

    E. E. Kummer, ¨Uber die Erg¨ anzungss¨ atze zu den allgemeinen Reciprocit¨ atsgesetzen, J. Reine Angew. Math.44(1852), 93–146

  2. [10]

    Kurz,Divisible codes, arXiv:2112.11763 (2021)

    S. Kurz,Divisible codes, arXiv:2112.11763 (2021)

    2021

  3. [11]

    I. N. Landjev,The geometric approach to linear codes, Finite Geometries, 2001, pp. 247–256

    2001

  4. [12]

    Maruta,The nonexistence of some ternary linear codes of dimension 6, Discrete Math.288(2004), no

    T. Maruta,The nonexistence of some ternary linear codes of dimension 6, Discrete Math.288(2004), no. 1-3, 125–133

    2004

  5. [13]

    Solomon and J

    G. Solomon and J. J. Stiffler,Algebraically punctured cyclic codes, Information and Control8(1965), 170–179

    1965

  6. [14]

    Z. H. Sun,The combinatorial sum P k≡r(modm) n k and its applications in number theory. III, Nanjing Daxue Xuebao Shuxue Bannian Kan12(1995), no. 1, 90–102

    1995

  7. [15]

    Z. W. Sun,On the sum P k≡r(modm) n k and related congruences, Israel J. Math.128 (2002), 135–156

    2002

  8. [16]

    Z. W. Sun and R. Tauraso,Congruences for sums of binomial coefficients, J. Number Theory126(2007), no. 2, 287–296

    2007

  9. [17]

    H. N. Ward,Divisible codes, Arch. Math. (Basel)36(1981), no. 6, 485–494

    1981

  10. [18]

    H. N. Ward,Weight polarization and divisibility, Discrete Math.83(1990), no. 2-3, 315–326

    1990

  11. [19]

    H. N. Ward,Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A83(1998), no. 1, 79–93

    1998

  12. [20]

    H. N. Ward,Divisible codes – a survey, Serdica Math. J.27(2001), no. 4, 263–278. 28

    2001

  13. [21]

    H. N. Ward,A sequence of unique quaternary Griesmer codes, Des. Codes Cryptogr. 33(2004), no. 1, 71–85

    2004

  14. [22]

    H. N. Ward and J. A. Wood,Characters and the equivalence of codes, J. Combin. Theory Ser. A73(1996), no. 2, 348–352. Department of Mathematics and National Center for Applied Mathematics Shenzhen, South- ern University of Science and Technology, Shenzhen 518055, China Email ad...

    1996

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