Perfect codes in weakly metric association schemes
Combinatorics
2026-01-21 v1 Information Theory
math.IT
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 -distance regular graphs ( Bernard et al 2025) is pointed out.
Cite
@article{arxiv.2601.12818,
title = {Perfect codes in weakly metric association schemes},
author = {Minjia Shi and Jing Wang and Patrick Solé},
journal= {arXiv preprint arXiv:2601.12818},
year = {2026}
}