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Perfect codes in weakly metric association schemes

arXiv CS Tuesday 02 June 2026, 04:00 UTC By Minjia Shi, Jing Wang, Patrick Sol\'e 1 min read
Key Points

Announce Type: replace-cross Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes.

arXiv:2601.12818v2 Announce Type: replace-cross Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes. A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of $m$-distance regular graphs ( Bernard et al 2025) is pointed out.
The Lloyd Theorem (ORG) Sol\'e (ORG) the Schwartz-Zippel Lemma (ORG) NRT (LOCATION) integer (ORG) \em (ORG) Bannai et al (PERSON) Bernard et al 2025 (PERSON)
Originally published by arXiv CS Read original →

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