arxiv: 2605.12148 · v1 · submitted 2026-05-12 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Closed Expressions for the Weight Distributions of Codes Associated with Perfect Codes

Denis Krotov, Minjia Shi, Tuvi Etzion, Wenhao Song

Pith reviewed 2026-05-13 04:38 UTC · model grok-4.3

classification 🧮 math.CO

keywords weight distributionsperfect codescovering codescompletely regular codescombinatorial counting1-perfect codesextended perfect codesdiamond codes

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The pith

Complete weight distributions are derived for five families of codes linked to perfect codes by elementary combinatorial methods.

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

This paper brings together five structures in coding theory for the first time: 1-perfect codes, their extensions, nearly perfect 1-covering codes and their extensions, plus diamond codes. It calculates the full weight distribution for each family. The calculations use basic counting arguments rather than heavy algebraic or other advanced tools. Knowing these distributions shows exactly how many codewords exist at each possible weight, which is useful for understanding the codes' structure and performance in detecting and correcting errors. The approach treats all five families uniformly because of their shared combinatorial properties.

Core claim

The paper shows that the weight distributions of 1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and diamond codes can all be expressed in closed form through direct combinatorial counting without the need for sophisticated mathematical machinery.

What carries the argument

The uniform application of combinatorial counting techniques to determine the number of codewords of each weight in these five code families.

Load-bearing premise

The five structures are assumed to be sufficiently related that the same combinatorial counting techniques apply uniformly to all of them without additional case-by-case adjustments.

What would settle it

Computing the weight distribution for a specific small-length code in one of the families, such as a diamond code of length 7, and finding it differs from the formula provided would disprove the general expression.

read the original abstract

Perfect codes are arguably the most fascinating structures in combinatorial coding theory, and their classification and weight distribution are of considerable interest. This classification also involves the analysis of some related structures. This paper considers five closely related structures, but all of them have never been tied together before. These structures are 1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and one family of completely regular codes (to be called diamond codes). The current work concentrates on the weight distributions of these five families of codes. In the past, some of these weight distributions were not computed, some required heavy tools, and for some only the weight enumerator was presented. We provide complete weight distributions for all five families using some methods that do not require any heavy tools.

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 supplies closed weight distributions for five related code families using elementary combinatorial counting.

read the letter

Hi colleague, The key takeaway from this paper is that it derives closed-form weight distributions for five interconnected families of codes in combinatorial coding theory: the 1-perfect codes, their extended versions, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and diamond codes. These had not all been linked in this way previously, and some distributions were either uncomputed or relied on more complex methods. What the paper does well is to use basic combinatorial counting to obtain these distributions uniformly across the families. It builds on the standard relationships, such as extensions by adding coordinates and coverings derived from Hamming spheres, to establish the necessary equivalences or bijections. This allows the same style of arguments to work for all five without needing heavy tools like those in some prior studies. The result is complete weight distributions presented explicitly, which fills gaps and makes the information more readily available. The soft spots are minor. The main one is that the uniformity of the method across all families, particularly for the diamond codes which are completely regular, might involve some specific adjustments. The abstract suggests the counting works directly from the code definitions, and there are no signs of circularity or free parameters. This work is primarily for specialists in coding theory, especially those interested in perfect codes and their variants. A reader who needs the weight distributions for these structures will get direct value from the closed expressions. It engages honestly with the existing literature by noting what was missing before and providing the new results. I think it deserves a serious referee because it contributes concrete, usable formulas in an established area. My recommendation is to send it to peer review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper derives closed-form weight distributions for five families of codes associated with perfect codes—1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and diamond codes—by applying uniform combinatorial counting arguments after establishing the necessary relations among the families, without relying on heavy machinery.

Significance. If the derivations are correct, the work provides a valuable unified treatment of these related structures, making previously unavailable or cumbersome weight distributions explicit and accessible. This strengthens the combinatorial toolkit for analyzing perfect and nearly perfect codes and their extensions.

minor comments (2)

[Abstract / Introduction] The abstract states that some weight distributions 'were not computed' or 'required heavy tools' previously; a brief table or paragraph in the introduction explicitly identifying which families fall into each category would strengthen the motivation.
[Introduction] The claim that the five structures 'have never been tied together before' would benefit from a short diagram or explicit list of the bijections/equivalences used to transfer the counting arguments across families.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its unified combinatorial approach, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation self-contained via direct combinatorial counting

full rationale

The paper states that it obtains complete weight distributions for the five families (1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and diamond codes) by applying uniform combinatorial counting techniques derived from the definitions of the codes and their standard relations (extensions, coverings, completely regular properties). No equations reduce a claimed prediction to a fitted parameter by construction, no self-definitional loops appear, and no load-bearing uniqueness theorems or ansatzes are imported solely via self-citation. The abstract and approach emphasize methods that avoid heavy tools, indicating the results follow from explicit counting arguments applied to the code structures themselves rather than from re-expressing inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard combinatorial properties of perfect codes and covering codes; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The standard definition and sphere-packing properties of 1-perfect codes hold over the relevant alphabets.
Invoked implicitly when relating the five code families.

pith-pipeline@v0.9.0 · 5439 in / 1048 out tokens · 39925 ms · 2026-05-13T04:38:17.356496+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear
We provide complete weight distributions for all five families using some methods that do not require any heavy tools.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear
The weight distribution of a perfect code ... Ai(C) = binom(n-1,i) + (n-1) Delta_i / n

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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