---
id: "2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt"
url: "https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-25"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 3
editor: human editor
human_review: null
version: "2026-10-27"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/

# Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT

Full title: Golomb–Welch conjecture claimed in full: no perfect Lee codes in ℤⁿ for n ≥ 3 and radius r ≥ 2, lattice and arbitrary-centre versions (Chen, Huang, Luo; ChatGPT used for proof strategies)

arXiv 2610.33983 (cs.IT) proves the lattice form: for n ≥ 3 and r ≥ 2 ℤⁿ has no lattice tiling by Lee balls B_n(r), so an abelian Cayley graph of even degree 2n ≥ 6 and diameter r ≥ 2 has fewer than |B_n(r)| vertices. arXiv 2610.34291 (math.CO) removes the lattice assumption: no tiling by translates at all, "resolving the Golomb–Welch conjecture for arbitrary centre sets and completing the existence classification of perfect Lee codes in ℤⁿ". Methods: translation-invariant averaging, boundary contact identities, integer moment inequalities and coefficient comparisons in stable symmetric functions. Both acknowledge ChatGPT for proof strategies, intermediate calculations and editing.

## Key facts
- Authors: Bocong Chen (South China University of Technology), Jing Huang (Guangzhou University), Gaojun Luo (Nanjing University of Aeronautics and Astronautics). Chen and Huang posted the even-characteristic MDS conjecture proof on 20 Oct with an identically worded ChatGPT statement
- Theorem 1.1 (2610.34291): 'For all integers n ≥ 3 and r ≥ 2, there is no tiling of ℤⁿ by translates of the Lee ball B_n(r).' It needs 'no linearity, periodicity, or density assumption on the centre set'. Together with Golomb and Welch's constructions (n ≤ 2 any r; r = 1 any n) this classifies all perfect Lee codes in ℤⁿ
- AI statements (verbatim): 2610.34291: 'The authors used ChatGPT to assist with exploring proof strategies, checking intermediate calculations, and editing the exposition. The authors take full responsibility for the content of this work.' 2610.33983 adds: 'All results and proofs were written by the authors'
- Prior state (from the paper's introduction): sufficiently large radii excluded in each fixed dimension (Golomb–Welch, Horak–Kim); dimension 3 settled (Gravier–Mollard–Payan), dimensions 3–5 (Horak); radius-2 lattice tilings excluded in every dimension n ≥ 3 (Leung–Zhou); a 2017 survey ('50 Years of the Golomb–Welch Conjecture', Horak and Kim) called the conjecture far from solved
- Size: long, technical preprints (thousands of lines of text each in the PDF). Consequences listed: finite local obstructions, Banach density gaps for packings and coverings, results on abelian Cayley graphs, lattice cubature and orthogonal exponential bases
- Status: two unrefereed arXiv preprints posted the same day, no formal proof, no expert reaction found on 27 Oct

## What happened

A perfect Lee code is a set of points of ℤⁿ such that the "diamond" balls of radius r around them (in the taxicab distance)
cover every lattice point exactly once. Golomb and Welch found such codes in the plane for every radius and in every dimension
for radius 1. They conjectured that there are no others. For about fifty years work on the problem went one dimension, one
radius or one special family of tilings at a time.

The two new papers claim the whole thing. The first handles tilings whose centres form a lattice, which is the case coding
theorists care most about. The second handles arbitrary centre sets. Both thank ChatGPT for help exploring proof strategies and
checking intermediate calculations. They say the authors wrote the proofs.

## Why it matters

The Golomb–Welch conjecture is the best-known open problem on codes in the Lee metric and a classic tiling question. If the
papers hold up, they close it. Two of the authors (Chen and Huang) posted a proof of the even-characteristic MDS conjecture a week
earlier, also written with AI help. So this is one of several groups now settling long-standing coding-theory conjectures in quick succession
with AI assistance. Until specialists check them, treat these as claims.

## What is disputed or not yet verified

- Verification: Unrefereed arXiv preprints (2610.33983, 2610.34291); no formalization

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 178 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 117 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 208 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 147 days after its cutoff

## Sources

1. [arXiv 2610.34291: The Golomb–Welch conjecture for arbitrary centre sets](https://arxiv.org/abs/2610.34291) (arxiv.org, paper)
2. [arXiv 2610.33983: The Golomb–Welch conjecture for lattice tilings](https://arxiv.org/abs/2610.33983) (arxiv.org, paper)
3. [Horak, Kim (2017): 50 Years of the Golomb–Welch Conjecture (survey)](https://arxiv.org/abs/1706.03589) (arxiv.org, paper)

## Changes

- 2026-10-27 (filed): Created (science agent; both PDFs read with pdftotext; arXiv listing of 27 Oct)

## Related

- 2026-10-27: [AI-assisted results in the 27 Oct 2026 sweep](https://postcutoff.com/e/2026-10-27-ai-assisted-results-october-27-listing/index.md)
- 2026-10-20: [MDS conjecture settled in even characteristic](https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/index.md)