Perfect and Quasi-Perfect Codes Under the $l_{p}$ Metric

A long-standing conjecture of Golomb and Welch, raised in 1970, states that there is no perfect <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> error correcting Lee code of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$n\geq 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$r>1$ </tex-math></inline-formula>. In this paper, we study perfect codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}^{n}$ </tex-math></inline-formula> under the <inline-formula> <tex-math notation="LaTeX">$l_{p}$ </tex-math></inline-formula> metric, where <inline-formula> <tex-math notation="LaTeX">$1\leq p<\infty $ </tex-math></inline-formula>. We show some nonexistence results of linear perfect <inline-formula> <tex-math notation="LaTeX">$l_{p}$ </tex-math></inline-formula> codes for <inline-formula> <tex-math notation="LaTeX">$p=1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$2\leq p<\infty $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$r=2^{1/p},3^{1/p}$ </tex-math></inline-formula>. We also give an algebraic construction of quasi-perfect <inline-formula> <tex-math notation="LaTeX">$l_{p}$ </tex-math></inline-formula> codes for <inline-formula> <tex-math notation="LaTeX">$p=1, r=2$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$2\leq p<\infty , r=2^{1/p}$ </tex-math></inline-formula>.