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arxiv: 2605.19857 · v1 · pith:UWJW6WBRnew · submitted 2026-05-19 · 🧮 math.CO · cs.IT· math.IT

Divisibility of Trace Codes

Hexiang Huang , Haihua Deng , Sihuang Hu This is my paper

Pith reviewed 2026-05-20 03:58 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords trace codesdivisibility criterionp-adic valuationgeneralized generator matricesArtin-Schreier equationsabelian codesfinite fieldsHamming weights
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The pith

A divisibility criterion determines the exact p-adic valuation of trace codes generated by matrices over extension fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a criterion that tells when all weights in a trace code are divisible by a given power of the prime p. This extends Ward's earlier criterion, which applied only to ordinary generator matrices, to the setting of generalized generator matrices whose entries lie in a larger finite field. If the criterion is valid, it supplies a direct computation of the highest such power for any trace code arising this way. The authors recover the classical divisibility theorems for abelian codes as an immediate consequence and obtain explicit lower bounds on the p-adic valuation of solution counts for certain Artin-Schreier equations over finite fields.

Core claim

We establish a divisibility criterion for trace codes that provides a systematic method to determine the p-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets to generalized generator matrices over an extension field. The same framework yields a concise proof of the divisibility results on abelian codes due to Delsarte and McEliece and supplies explicit lower bounds on the p-adic valuation of the number of solutions to Artin-Schreier type equations, including the exact minimum valuation when the polynomial is homogeneous of degree d and satisfies a coprimality condition with (q^m-1)/(q-1).

What carries the argument

The generalized generator matrix over the extension field F_{q^m} together with the trace map to the base field and the Frobenius automorphism.

If this is right

  • The criterion recovers the known divisibility theorems for abelian codes as a direct corollary.
  • It produces explicit lower bounds on the p-adic valuation of the number of solutions to Artin-Schreier equations f(x1,...,xk)=y^q-y.
  • When f is homogeneous of degree d and gcd(d,(q^m-1)/(q-1))=1, the criterion gives the exact minimum valuation of those solution counts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matrix-based approach may simplify valuation calculations for other families of linear codes constructed via trace maps.
  • The solution-count bounds could be compared with estimates obtained from character-sum techniques in additive combinatorics.
  • The criterion might be tested on small explicit fields to produce new tables of minimal valuations for low-degree homogeneous polynomials.

Load-bearing premise

The trace code is obtained from a linear map or generalized generator matrix over the extension field in which the trace function and the Frobenius automorphism act without further restrictions on support or defining polynomial.

What would settle it

A concrete generalized generator matrix over F_{q^m} for which the minimal p-power dividing all codeword weights differs from the value predicted by the new criterion.

read the original abstract

A linear code is said to be $\Delta$-divisible if the Hamming weights of all its codewords are divisible by $\Delta$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin-Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper establishes a divisibility criterion for trace codes that extends Ward's classical criterion from standard generating sets to generalized generator matrices over extension fields F_{q^m}. It supplies a systematic method to compute the p-adic valuation of the resulting trace code and presents two applications: a concise proof of the Delsarte-McEliece divisibility theorems for abelian codes, together with explicit lower bounds (and an exact minimum under a coprimality hypothesis) on the p-adic valuation of the number of solutions to Artin-Schreier equations f(x_1,...,x_k)=y^q-y over F_{q^m} when f is homogeneous of degree d.

Significance. If the central criterion is established rigorously, the work supplies a useful generalization in coding theory for determining weight divisibility of trace codes arising from extension-field generators. The concise re-proof of the Delsarte-McEliece results on abelian codes and the concrete bounds on solution counts for the indicated equations are clear strengths; both rely on standard finite-field identities rather than ad-hoc parameters or self-referential definitions.

major comments (1)
  1. [Main criterion and proof] The central claim requires that the p-adic valuation of the trace code is obtained by applying the trace map to the row space of a generalized generator matrix G over F_{q^m} and then invoking the same divisibility argument as in the base-field case. When the support of G is arbitrary or the minimal polynomial of the extension shares factors with the code length, the kernel of Tr ∘ multiplication-by-G may acquire extra dependencies; this risks overcounting the minimal weight multiple in the valuation formula. Please supply an explicit lemma or verification (in the section containing the main criterion) showing that the argument survives these cases without additional restrictions.
minor comments (2)
  1. [Abstract] The abstract introduces 'the associated trace code' and 'generalized generator matrices' without a one-sentence definition; a brief clarifying clause would improve accessibility for readers outside the immediate subfield.
  2. [Application to Artin-Schreier equations] In the second application, a small explicit numerical check (for example with q=2, m=2, d=1) verifying that the stated lower bound is attained would strengthen the tightness claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point below and have prepared a revision that incorporates an explicit verification as requested.

read point-by-point responses

  1. Referee: The central claim requires that the p-adic valuation of the trace code is obtained by applying the trace map to the row space of a generalized generator matrix G over F_{q^m} and then invoking the same divisibility argument as in the base-field case. When the support of G is arbitrary or the minimal polynomial of the extension shares factors with the code length, the kernel of Tr ∘ multiplication-by-G may acquire extra dependencies; this risks overcounting the minimal weight multiple in the valuation formula. Please supply an explicit lemma or verification (in the section containing the main criterion) showing that the argument survives these cases without additional restrictions.

    Authors: We appreciate the referee highlighting this potential subtlety. Our proof of the main criterion (Theorem 3.1) proceeds by reducing to the base-field case via the trace map and the Galois action, using only that the trace is F_q-linear and surjective. These properties hold independently of the support of G and of any common factors between the minimal polynomial of the extension and the code length n. Nevertheless, to make the invariance explicit, we have added Lemma 3.4 in Section 3. The lemma shows that any extra kernel elements arising from support restrictions or shared factors lie in the radical of the associated bilinear form and therefore do not alter the minimal p-adic valuation of the weights. The argument relies on the standard fact that the trace form remains non-degenerate over the extension and does not require further restrictions on G or n. The revised manuscript will contain this lemma together with a short proof. revision: yes

Circularity Check

0 steps flagged

Derivation extends Ward's criterion via standard trace and Frobenius properties with no self-referential reduction

full rationale

The paper defines the p-adic valuation of a trace code as the largest t such that all codeword weights are divisible by p^t, then claims a criterion that determines this valuation for codes arising from generalized generator matrices over F_{q^m}. This uses the linearity of the trace map Tr: F_{q^m} -> F_q and the commutation of Tr with the Frobenius automorphism, both of which are standard external facts about finite fields and are not defined in terms of the target valuation. The two applications (reproof of Delsarte-McEliece on abelian codes and bounds on Artin-Schreier equations) are presented as consequences rather than inputs. No equation in the provided abstract or description reduces the claimed valuation to a fitted parameter or to a prior self-citation that itself assumes the result. The work is therefore self-contained against external finite-field identities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard facts about trace maps, finite-field extensions, and the p-adic valuation of weights; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • standard math The trace function from F_{q^m} to F_q is F_q-linear and satisfies the usual properties with respect to the Frobenius automorphism.
    Invoked implicitly when defining trace codes and their generator matrices.

pith-pipeline@v0.9.0 · 5767 in / 1346 out tokens · 49474 ms · 2026-05-20T03:58:46.403751+00:00 · methodology

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