Perfect codes in weakly metric association schemes
Minjia Shi, Jing Wang, Patrick Solé
TL;DR
The work addresses the nonexistence of $e$-perfect codes in four metric contexts by fusing Lloyd-type necessary conditions with the Schwartz-Zippel bound within the framework of polynomial weakly metric association schemes (WMAS). Central to the approach is the dispersion function $oldsymbol{ abla}$ and the Lloyd polynomial $oldsymbol{ abla}_e$, whose multivariate polynomial nature in an $r$-polynomial WMAS enables a tight contradiction when $oldsymbol{ abla}(e)-1 > e|S|^{r-1}$. The master theorem yields broad nonexistence results for Lee, NRT, sum-rank, and mixed-alphabet codes, supported by asymptotic partition counts; the results are complemented by numerical illustrations and explicit corollaries. The paper also clarifies connections to multivariate $P$-polynomial schemes and $m$-distance regular graphs, providing a path for refined analysis beyond the current scope and identifying regimes where the method does or does not apply.
Abstract
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.