The Lloyd Theorem of (Solé, 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.

翻译：结合(Solé, 1989)的Lloyd定理与理论计算机科学中的Schwartz-Zippel引理，推导出Lee度量、NRT度量、混合汉明度量以及和秩距离下完美码的非存在性结果。证明基于整数划分的渐近计数。该框架基于新的{\em 多项式}弱度量关联方案概念。本文指出了这一概念与(Bannai等人, 2025)提出的多元P-多项式方案理论以及(Bernard等人, 2025)提出的$m$-距离正则图理论之间的联系。