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 r error correcting Lee code of length n for n ≥ 3 and r > 1. In this paper, we study perfect codes in Zn under the l p metric, where 1 <; p <; ∞. We show some nonexistence results of linear perfect l p codes for p = 1 and 2 <; 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 <; p <; ∞, r = 21/p.