Quasi-perfect codes in the $\ell_p$ metric

TL;DR

This paper investigates the existence and properties of quasi-perfect codes in the integer lattice under the _p metric, providing computational classifications for specific dimensions and exploring generalized notions of code perfection.

Contribution

It offers a comprehensive computational classification of linear quasi-perfect codes in low dimensions for the _p metric and introduces a generalized concept of imperfection in coding theory.

Findings

Identified all radii for linear quasi-perfect codes in _2 for dimensions 2 and 3.

Numerical results on codes with minimal degree of imperfection.

Extended the concept of code perfection to a broader class of codes.

Abstract

We consider quasi-perfect codes in $\mathbb{Z}^n$ over the $\ell_p$ metric, $2 \leq p < \infty$. Through a computational approach, we determine all radii for which there are linear quasi-perfect codes for $p = 2$ and $n = 2, 3$. Moreover, we study codes with a certain \textit{degree of imperfection}, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented.