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Golomb-Welch Conjecture

The Golomb-Welch conjecture asserts that, for integers n>=3 and r>=2, the integer lattice Z^n cannot be tiled by translates of the radius-r ball in the Lee metric (Golomb and Welch 1970). Equivalently, there is no perfect Lee code with those parameters.

Chen et al. (2026b) reported a proof for lattice tilings, in which the translation centers form a lattice. Chen et al. (2026a) reported an extension to arbitrary translation centers. These are companion claims by the same authors. Independent verification of the full arguments has not been established, so they are not treated here as an accepted resolution. Chen et al. (2026ab) report AI assistance with mathematical proof development and verification.

See also

Lee Metric, Perfect Code

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References

Chen, B.; Huang, J.; and Luo, G. "The Golomb-Welch Conjecture for Arbitrary Centre Sets." 25 Oct 2026a. https://arxiv.org/abs/2610.34291.Chen, B.; Huang, J.; and Luo, G. "The Golomb-Welch Conjecture for Lattice Tilings." 25 Oct 2026b. https://arxiv.org/abs/2610.33983.Golomb, S. W. and Welch, L. R. "Perfect Codes in the Lee Metric and the Packing of Polyominoes." SIAM J. Appl. Math. 18, 302-317, 1970. https://doi.org/10.1137/0118025.

Cite this as:

Weisstein, Eric W. "Golomb-Welch Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Golomb-WelchConjecture.html

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