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

Abstract

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