Perfect codes in weakly metric association schemes

TL;DR

This paper establishes non-existence results for perfect codes across various metrics using a novel framework of polynomial weakly metric association schemes, combining combinatorial and algebraic techniques.

Contribution

It introduces the concept of polynomial weakly metric association schemes and connects it to recent theories, providing new tools for analyzing perfect codes.

Findings

Non-existence of perfect codes in Lee, NRT, mixed Hamming, and sum-rank metrics.

Application of the Lloyd Theorem and Schwartz-Zippel Lemma to coding theory.

Asymptotic enumeration of integer partitions underpins the proofs.

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.