Quasi-perfect codes in the ${\ell }_p$ metric

Abstract

We consider quasi-perfect codes in ${\mathbb{Z}}^n$ over the ${\ell }_p$ metric, $2\le p<\infty$. 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.

15 pages, 6 figures