arxiv: 2605.10594 · v1 · submitted 2026-05-11 · 💻 cs.IT · math.IT

Recognition: 1 theorem link

· Lean Theorem

Weight distributions of cosets of weight 2 of the generalized doubly extended Reed-Solomon codes

Alexander A. Davydov , Stefano Marcugini , Fernanda Pambianco

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classification 💻 cs.IT math.IT

keywords generalized Reed-Solomon codescoset weight distributionsdoubly extended codesfinite fieldscombinatorial counting problemsweight enumeratorsReed-Solomon codesCase S

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The pith

If q-1 and d-2 are coprime then all weight-2 cosets of generalized doubly extended Reed-Solomon codes have the same weight distribution

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

The paper proves that for generalized doubly extended Reed-Solomon codes with minimum distance d at least 5 over the field with q elements, the coprimeness of q-1 and d-2 forces every coset of weight 2 to share one common weight distribution. This uniform distribution was already known, so the result gives an explicit answer for all such cosets whenever the gcd condition holds. The authors also introduce two equivalent open counting problems over finite fields and over the ring of integers modulo q-1, show that their solutions determine the distributions in the remaining cases, and solve the additive version for many concrete pairs of parameters.

Core claim

For a generalized doubly extended Reed-Solomon code of minimum distance d >= 5 over F_q, if q-1 and d-2 are coprime then Case S holds: the weight distribution is identical for every coset of weight 2. In this situation the code is 2-regular. The complementary Case NS reduces exactly to solving the counting problems A_{q,mu}^x and A_{R,mu}^+ with mu = d-2 and R = q-1, where the ring problem asks for the number of mu-tuples of distinct elements summing to a given lambda.

What carries the argument

The coprimeness of q-1 and d-2, which collapses all weight-2 cosets into a single equivalence class whose weight distribution is already known

If this is right

When q-1 and d-2 are coprime the weight distribution of every weight-2 coset is the known uniform one
The code satisfies the 2-regular property
The weight distributions in Case NS are determined by the number of distinct mu-tuples summing to each lambda in Z_{q-1}
Solutions of the ring counting problem for concrete R and mu give the coset weight distributions for the corresponding q = R+1 and d = mu+2

Where Pith is reading between the lines

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

The two new counting problems are of independent combinatorial interest and may admit further closed-form solutions for additional parameter pairs
The reduction shows that certain questions about cosets of algebraic codes translate directly into elementary additive or multiplicative counting tasks over rings and fields
Completing the solution of the counting problems for all mu and R would give the weight distributions for every such code without any coprimeness restriction

Load-bearing premise

The weight distribution problem for weight-2 cosets of these codes reduces exactly to the stated combinatorial counting problems over the multiplicative group of the field and over the ring of integers modulo q-1.

What would settle it

Direct computation of the weight distributions of two distinct weight-2 cosets for a small pair q,d satisfying gcd(q-1,d-2)=1 and checking whether the two distributions differ

read the original abstract

We consider the weight distributions of the cosets of weight 2 of the generalized $[q+1,q+2-d,d]_q$ doubly extended Reed-Solomon codes (GDRS) of minimum distance $d\ge5$, over the finite field $\mathbb{F}_q$ with $q$ elements. For a GDRS code, we say that Case S occurs if the weight distribution for all cosets of weight 2 is the same or otherwise, Case NS occurs. For Case S, the weight distribution is known; however, any sufficient condition for the occurrence of Case S remained an open problem. We prove that if $q-1$ and $d-2$ are coprime then Case S holds, i.e. the problem is solved. Furthermore, we note that in Case S, the GDRS code is 2-regular. Also, we introduce two new open equivalent combinatorial problems for finite fields $\mathbb{F}_q$ (Problem $A_{q,\mu}^\times$) and for rings $\mathbb{Z}_\mathfrak{R}$ of integers modulo $\mathfrak{R}$ (Problem $A_{\mathfrak{R},\mu}^+$), where $\mu$ is a parameter. In particular, Problem $A_{\mathfrak{R},\mu}^+$ is as follows: for each element $\lambda$ of $\mathbb{Z}_\mathfrak{R}$, determine the number of all possible $\mu$-tuples $\{\lambda_1,\lambda_2,\ldots,\lambda_{\mu}\}$, each of which consists of $\mu$ distinct elements $\lambda_j$ of $\mathbb{Z}_\mathfrak{R}$ such that their sum in $\mathbb{Z}_\mathfrak{R}$ is equal to $\lambda$. Open Problems $A_{q,\mu}^\times$ and $A_{\mathfrak{R},\mu}^+$ are interesting in their own right and, moreover, we proved that their solutions allow us to obtain the weight distributions for Case NS, taking $\mu=d-2$ and $\mathfrak{R}=q-1$. To solve Problem $A_{\mathfrak{R},\mu}^+$, we found a universal method, connected with the values of $\mathfrak{R}$ and $\mu$, using orbits of elements in $\mathbb{Z}_\mathfrak{R}$ and then we solved the problem for many pairs $\mathfrak{R},\mu$, obtaining the needed weight distributions for the corresponding pairs $q=\mathfrak{R}+1,d=\mu+2$.

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.

Desk Editor's Note private letter to a colleague

The paper proves that gcd(q-1, d-2)=1 implies uniform weight distributions for all weight-2 cosets of these GDRS codes.

read the letter

The main result is a proof that when q-1 and d-2 are coprime, the weight-2 cosets of generalized doubly extended Reed-Solomon codes all share the same weight distribution. This settles the open problem mentioned in the abstract. The authors reduce the coset weight problem to counting the number of mu-tuples of distinct elements summing to a given lambda in F_q^* or equivalently in Z_{q-1}. Under the coprime hypothesis, multiplication by mu is a bijection modulo q-1, making the count independent of lambda. This is a direct and elementary argument using the standard description of RS syndromes and basic properties of cyclic groups. They also show the code is then 2-regular. They introduce two equivalent open combinatorial problems and solve the ring version for many specific (R, mu) pairs via an orbit-based counting method. This gives explicit weight distributions for the non-uniform cases when the gcd condition fails. The paper does a good job laying out the reduction and handling the uniform case cleanly. The combinatorial approach is reproducible and doesn't involve any fitting or circular reasoning. The stress-test note confirms the argument uses only standard tools without hidden assumptions. The limitation is that the general counting problem is left open, so the result is complete only for the coprime parameters. For other cases, users get solutions only for the pairs the authors computed. No major flaws in the presented work, but the scope is narrower than a full resolution. This is aimed at coding theorists working on MDS codes and exact performance analysis. Someone needing closed-form coset weights for GDRS codes will get value from the solved cases and the condition. It deserves a serious referee because it answers an open question with verifiable math. I recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a sufficient condition for the weight distributions of all weight-2 cosets of generalized doubly extended Reed-Solomon codes to be identical. Specifically, for a [q+1, q+2-d, d]_q GDRS code with d ≥ 5, if gcd(q-1, d-2) = 1, then Case S holds: every weight-2 coset has the same weight enumerator. The proof proceeds by reducing the syndrome equations to the problem of counting μ-tuples of distinct elements in F_q^* summing to a given λ (μ = d-2), or equivalently counting μ-tuples in Z_{q-1} summing to λ mod (q-1). Under the coprimality hypothesis, multiplication by μ is bijective on Z_{q-1}, making the number of solutions independent of λ. The authors additionally pose two equivalent open combinatorial problems (A_{q,μ}^× and A_{R,μ}^+) and provide explicit solutions for numerous (R, μ) pairs using an orbit-based counting method, thereby determining the weight distributions in the complementary Case NS. They also note that Case S implies the code is 2-regular.

Significance. If the central argument is correct, the result supplies the first explicit sufficient condition resolving the weight-2 coset distribution problem for a large class of GDRS codes. The reduction to additive combinatorics over the cyclic group Z_R is elementary and leverages only standard properties of Reed-Solomon locators and group homomorphisms. The introduction of Problems A_{q,μ}^× and A_{R,μ}^+ as independently interesting questions, together with the concrete solutions for many parameter pairs, adds value beyond the coding-theoretic application. The observation that Case S implies 2-regularity of the code is a pleasant corollary. These contributions advance the understanding of coset weight enumerators for algebraic codes.

minor comments (3)

[Abstract] Abstract: the phrasing 'i.e. the problem is solved' is imprecise; the text should clarify that only the coprime case receives a complete resolution while the general (non-coprime) case remains open and is addressed only for specific parameter pairs.
The claim that the GDRS code is 2-regular under Case S is stated without derivation or reference; a short paragraph recalling the definition of 2-regularity and showing why identical weight-2 coset enumerators imply it would strengthen the presentation.
In the section introducing Problems A_{q,μ}^× and A_{R,μ}^+, the equivalence via the discrete logarithm should be made fully explicit (including the precise mapping from F_q^* to Z_{q-1}) rather than asserted at a high level.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for recognizing the significance of the sufficient condition for Case S, the introduction of the open combinatorial problems, and the corollary on 2-regularity. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper reduces the weight-2 coset weight distribution of GDRS codes to the standard locator-polynomial syndrome equations, which in turn reduce exactly to the new combinatorial counting problems A_{q,μ}^× and A_{R,μ}^+ with μ = d-2 and R = q-1. Under the hypothesis gcd(q-1, d-2) = 1 the authors prove independence of the target sum λ by the bijectivity of multiplication by μ modulo R, an elementary fact from group theory on Z_R that does not rely on any fitted parameters, prior self-citations, or redefinition of the target quantity. The combinatorial problems are explicitly introduced as open and solved only for the complementary NS cases; the coprime Case S result stands independently. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs in the central chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the equivalence between coset weight distributions and the two new counting problems, plus standard facts about finite fields and Reed-Solomon codes; no free parameters or invented entities are introduced.

axioms (1)

domain assumption Standard algebraic properties of generalized doubly extended Reed-Solomon codes over finite fields F_q
The paper assumes the usual definition and minimum-distance properties of GDRS codes with d >= 5.

pith-pipeline@v0.9.0 · 5764 in / 1163 out tokens · 47099 ms · 2026-05-12T04:39:04.730845+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean, IndisputableMonolith/Foundation/AlexanderDuality.lean, IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction, J_uniquely_calibrated_via_higher_derivative unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that if q-1 and d-2 are coprime then Case S holds... Open Problems A_{q,μ}^× and A_{R,μ}^+ ... using orbits of elements in Z_R

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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