Quasi-perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

TL;DR

This paper constructs 2-quasi-perfect Lee codes over certain prime fields with arbitrarily large dimensions, providing new insights into Lee code structures and their relation to Ramanujan graphs.

Contribution

The paper introduces a novel construction of 2-quasi-perfect Lee codes for primes satisfying specific congruences, applicable to arbitrarily large dimensions, and explores their connection to Ramanujan graphs.

Findings

Constructed 2-quasi-perfect Lee codes over $ ext{Z}_p^n$ for primes $p eq 0 mod 12$

Codes are close to perfect, supporting the conjecture on non-existence of perfect Lee codes in higher dimensions

Related graphs are Ramanujan, indicating deep links between coding theory and graph theory

Abstract

A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension $n\geq 3$ and radius $r\geq 2$. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.