The Golomb--Welch conjecture for lattice tilings

Bocong Chen, Jing Huang, Gaojun Luo

Oct 25, 2026arXiv:2610.33983v1

cs.IT

Scorecard· 16/16

8.5/10 impact

Claims to resolve the lattice Golomb–Welch conjecture for all n≥3, r≥2, a 55-year-old problem; discounted for verification difficulty and a patchwork proof.

Abstract

We prove the Golomb--Welch conjecture for lattice tilings: for every $n\ge3$ and $r\ge2$ , the integer lattice $Z^{n}$ admits no lattice tiling by Lee balls $B_{n} (r)$ of radius $r$ . Together with the classical constructions for $n\le2$ and for $r = 1$ , this determines all parameters for which a lattice Lee tiling exists. Equivalently, we show that every abelian Cayley graph of even degree $2n\ge6$ and diameter $r\ge2$ has strictly fewer than $∣ B_{n} (r) ∣$ vertices.

AI Impact Assessments

(1 model)

Core Contribution. The paper claims a full proof of the lattice case of the Golomb–Welch conjecture: for all n≥3 and r≥2, Z^n has no lattice tiling by Lee balls B_n(r). Equivalently, no even-degree abelian Cayley graph of degree 2n≥6 and diameter r≥2 attains the Dougherty–Faber Moore bound. The conjecture dates to 1968–70 and has resisted many partial attacks: geometric large-radius bounds (Golomb–Welch, Post, Lepistö, Horak–Kim), small-dimension results (n=3,4,5), r=2 for all n (Leung–Zhou), and arithmetic or group-ring results in sparse infinite families. A uniform resolution over all (n,r) would be a landmark in coding theory, tiling theory and the degree–diameter problem.

The strategy is a case analysis on the remainder c in 2n=(r+1)q+c. The tools are:

a signed-orbit convolution obstruction (a four-orbit weighted inequality) for n up to about r²/2;

boundary-fiber partition identities with moment bounds;

scalar-kernel collision estimates;

character bounds via a Perron-root comparison of a nonnegative matrix;

prime-quotient and congruence arguments for finitely many leftover parameters.

Methodological Rigor. The paper is long and dense, and its structure is plausible: the case split exhausts 0≤c≤r, and the finite exceptions are enumerated. Several things warrant caution:

Many steps rest on heavy, hand-verified polynomial and rational inequalities (Bernstein-basis expansions, large tables of margins, base-p expansions). These are difficult to audit, and I could not check them.

Several lemmas depend on exact tabulated data, such as the (a_k, δ_k) certificates and the congruence tables. Any single error would leave a gap.

The exceptional finite cases lean on external results (Horak n=3–5, Leung–Zhou r=2, Horak–Kim, Zhang–Zhou Cor. 4.5 for (3,8), Xiao–Zhou for small ball volumes). Citing those results is reasonable, but a few are very recent preprints. The reference list also contains unusual identifiers (arXiv:2610.xxxx, "Hexagon"), which is odd.

The acknowledgments state that ChatGPT was used to explore proof strategies and check calculations. This is a mild reliability flag for a proof of this length.

The proof is highly fragmented: many ad hoc thresholds (N_r = r(r+1)/2+2, q≤21, n<9r) and a case-by-case endgame. This pattern is where subtle errors typically hide. That said, individual lemmas are written with explicit derivations, and the logical dependencies are traceable.

Potential Impact. If correct, the result would be among the most significant in the area. It would settle a 55-year-old conjecture in its lattice form, determine exactly when linear perfect Lee codes exist, and show that the abelian Cayley Moore bound is unattainable for diameter ≥2 and degree ≥6. The new methods (boundary-fiber identities, shadow-counting with Matolcsi–Ruzsa, and character/Perron comparisons) could be reused for related problems: non-lattice Golomb–Welch, quasi-perfect codes, and other metric tilings. The non-lattice case remains open, which limits the claim's reach. The paper itself notes that the exact maximum order of AC(2n,r) is a separate problem.

Timeliness. The conjecture has recently seen a burst of algebraic progress (Zhang–Zhou, Leung–Zhou, Xiao–Zhou), and this paper claims to close the gap. The result is highly relevant to the coding-theory and combinatorial-geometry communities.

Strengths.

It resolves a famous open problem uniformly, with no arithmetic hypotheses on |B_n(r)|.

It unifies earlier threads (group rings, character sums, geometric bounds).

Most claims are accompanied by explicit verification.

Limitations.

The proof is extremely long and technical, which makes independent verification hard.

Heavy reliance on computed tables and many exceptional-case patches.

It covers only lattice tilings.

It depends on very recent external preprints.

The exposition is compressed, with notation-heavy statements and few intuitions for why the key inequalities should hold.

Assessment. The claimed result is of first-rank significance. Rigor and verifiability are good in form but difficult to confirm given the proof's length and patchwork nature, and the AI-assisted workflow is a mild concern. I discount the expected impact somewhat for the risk that the proof contains an undetected gap, but the expected value remains very high.

Rating:8.5/ 10

Significance 9Rigor 6.5Novelty 7.5Clarity 5

Generated Oct 27, 2026

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Scorecard· 16/16

8.5/10 impact

Claims to resolve the lattice Golomb–Welch conjecture for all n≥3, r≥2, a 55-year-old problem; discounted for verification difficulty and a patchwork proof.