On some perfect codes with respect to Lee metric

Abstract

In this paper, we obtain bounds on the number of parity check digits for Lee weight codes correcting errors of Lee weight 1, errors of Lee weight 2 or less, errors of Lee weight 3 or less and errors of Lee weight 4 or less over Z q ( q ⩾ 5, a prime) respectively. We also examine these bounds with equality to check for perfect codes and have shown the existence of perfect codes correcting errors of Lee weight 1 over Z 5 and perfect codes correcting errors of Lee weight 2 or less over Z 13. We have also shown the nonexistence of perfect codes correcting errors of Lee weight 2 or less over Z q when q = 4 n + 3 ( q prime) and correcting errors of Lee weight 3 or less and errors of Lee weight 4 or less over Z q (5 ⩽ q ⩽ 97, a prime). We further conjecture that there does not exist a perfect code correcting errors of Lee weight t or less ( t ⩾ 3) over Z q ( q ⩾ 5, a prime).