Erratum to "Lattice constellation and codes from quadratic number fields" [IEEE Trans. Inform. Theory, vol. 47, No. 4, May. 2001]

TL;DR

This paper corrects a mistake in a previously published article's metric, redefining it to enable proper encoding, decoding, and the existence of 1-error correcting perfect codes.

Contribution

It identifies and corrects an error in the original metric, establishing a valid metric and demonstrating the existence of new perfect codes under it.

Findings

The original metric was incorrect and invalidated previous procedures.

A new proper metric is defined for the codes.

Existence of 1-error correcting perfect codes under the new metric.

Abstract

We correct a partial mistake for a metric presented in the article "Lattice constellation and codes from quadratic number fields" [IEEE Trans. Inform. Theory, vol. 47, No. 4, May. 2001]. We show that the metric defined in the article is not true, therefore, this brings about to destroy the encoding and decoding procedures. Also, we define a proper metric for some codes defined in the article and show that there exist some $1-$error correcting perfect codes with respect to this new metric.