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

We consider quasi-perfect codes in ${\mathbb{Z}}^n$ over the ${\mathrm{\ell }}_p$ metric, $2\le p<\mathrm{\infty }$. 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 $\mathit{\text{degree}}\mathit{\text{of}}\mathit{\text{imperfection}}$, a notion that generalizes the quasi-perfect codes. Numerical results concerning the codes with the smallest degree of imperfection are presented.