Perfect Mannheim, Lipschitz and Hurwitz weight codes

Abstract

The set of residue classes modulo an element $\pi$ in the rings of Gaussian integers, Lipschitz integers and Hurwitz integers, respectively, are used as alphabets to form the words of error correcting codes. An error occurs as the addition of an element in a set $E\cal{$} to the letter in one of the positions of a word. If $E\cal{$} is the group of units in the original rings, then we obtain the Mannheim, Lipschitz and Hurwitz metrics, respectively. Some new perfect 1-error-correcting codes in these metrics are constructed. The existence of perfect 2-error-correcting codes is investigated by a computer search.