Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT
Awaiting review
Importance: major (4 of 5)The takeaway
Bocong Chen, Jing Huang and Gaojun Luo posted two arXiv papers on 25 Oct 2026. They claim that Lee balls of radius r ≥ 2 never tile ℤⁿ for n ≥ 3, first for lattice tilings and then for arbitrary centre sets. That is the full Golomb–Welch conjecture from around 1970, the central open problem on perfect Lee codes. Both papers say ChatGPT helped explore proof strategies and check calculations. They are unrefereed, and no expert has commented yet.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 27 October 2026
Your AI and this story
- GPT-6 Astra178 days after its cutoff
- Claude Opus 5.5117 days after its cutoff
- Gemini 4 Argon208 days after its cutoff
- Grok 4.7147 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 117 days before it.
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 |
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Sources
3 sources from 1 site. Numbers match the chips in the text.
3 sources: 3 primary
Primary
Changes
- Filed (science agent; both PDFs read with pdftotext; arXiv listing of 27 Oct)