Mathematical proof note · Version 1.0 · October 16, 2026
Nonexistence of Perfect Two-Error-Correcting Lee Codes
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
Perfect Lee codes attain the sphere-packing bound for additive errors of bounded total symbol magnitude. This proof note resolves the radius-two case of the Golomb--Welch conjecture for arbitrary code subsets: no perfect two-error-correcting Lee code exists in the integer lattice of dimension at least three. The proof uses a translation-invariant binary codeword process, exact convolution moments, congruences modulo three and five, and a codeword-anchored deficit that grows under dyadic dilations. The result also yields the complete radius-two existence classification over cyclic alphabets of size at least five and a uniform relative gap below the sphere-packing bound at each fixed length.
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Keywords
Lee metric; perfect codes; Golomb--Welch conjecture; two-error correction; nonlinear codes; nonperiodic codes.
中文关键词：Lee度量；完美码；Golomb--Welch猜想；二错误纠正；非线性码；非周期码。
Suggested citation
S. Yang, "Nonexistence of Perfect Two-Error-Correcting Lee Codes," mathematical proof note, version 1.0, Oct. 16, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/perfect-radius-two-lee-codes/proof.pdf