Mathematical proof note · Version 1.0 · October 8, 2026
The Euclidean-Weight GCD of Self-Dual Codes over Odd Residue Rings
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
For a self-dual code over an integer residue ring, self-orthogonality makes the modulus a common divisor of the Euclidean weights. We prove that, for every odd modulus m ≥ 3, the greatest common divisor of all nonzero Euclidean weights is exactly m. The proof applies to nonfree codes as well as free codes. A finite-field character coefficient isolates coordinate lines and excludes every prime different from the characteristic. Residue–top-torsion duality transfers this exclusion to prime-power rings. A saturation argument and scalar-orbit sum rule out one additional factor of the characteristic. Exact scaling of balanced representatives then assembles the result over composite odd moduli.
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Keywords
self-dual codes; Euclidean weights; odd residue rings; greatest common divisor; nonfree codes.
中文关键词：自对偶码；欧几里得重量；奇数剩余类环；最大公约数；非自由码。
Suggested citation
S. Yang, "The Euclidean-Weight GCD of Self-Dual Codes over Odd Residue Rings," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/self-dual-euclidean-weight-gcd/proof.pdf