A perfect Lee code PLC(e, n) is a subset C ⊆ ℤ^n such that every point of ℤ^n is within Lee distance e of exactly one codeword. Question: classify all (e, n) pairs admitting perfect Lee codes. Known: e = 1 exists for all n; e = 2 exists for n ≡ 0 or 3 mod 4 under divisibility conditions; e ≥ 3 conjectured nonexistent for n ≥ 3 (Golomb–Welch).
Golomb and Welch conjectured in 1970 that PLC(e,n) doesn't exist for n ≥ 3 and e ≥ 2, based on density arguments. Partial results: Post proved nonexistence for e = 2, n ≡ 1, 2 mod 4; Kim made significant progress using tiling theory. The problem connects to lattice tilings, quasi-perfect codes, and the algebraic structure of Lee spheres (which are NOT polyominoes in n ≥ 3, unlike their Hamming-metric analogues).
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