Diameter Perfect Lee Codes

TL;DR

This paper investigates the existence and enumeration of diameter perfect Lee codes, providing algebraic constructions, characterizing when they exist, and demonstrating their abundance for certain parameters, contrasting with the uniqueness of perfect error-correcting Lee codes.

Contribution

It determines all q for which linear diameter-4 perfect Lee codes exist and shows uncountably many such codes for each n ≥ 3, using an algebraic group homomorphism-based construction.

Findings

Characterization of q for linear diameter-4 perfect Lee codes

Existence of uncountably many diameter-4 perfect Lee codes for each n ≥ 3

An algebraic construction enabling efficient decoding algorithms

Abstract

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper.