On some perfect codes with respect to Lee metric Original Research Article

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 Zq (q greater-or-equal, slanted 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 Z5 and perfect codes correcting errors of Lee weight 2 or less over Z13. We have also shown the nonexistence of perfect codes correcting errors of Lee weight 2 or less over Zq when q = 4n + 3 (q prime) and correcting errors of Lee weight 3 or less and errors of Lee weight 4 or less over Zq (5 less-than-or-equals, slant q less-than-or-equals, slant 97, a prime). We further conjecture that there does not exist a perfect code correcting errors of Lee weight t or less (t greater-or-equal, slanted 3) over Zq (q greater-or-equal, slanted 5, a prime).