TL;DR
This paper introduces new product constructions for perfect Lee and diameter perfect Lee codes, resulting in many nonlinear codes and discussing related open problems in the field.
Contribution
It presents novel product constructions for perfect Lee and diameter perfect Lee codes, expanding the known classes of nonlinear codes in these metrics.
Findings
Produced numerous nonlinear perfect Lee codes.
Developed new diameter perfect Lee codes.
Provided a survey of related open problems.
Abstract
A well known conjecture of Golomb and Welch is that the only nontrivial perfect codes in the Lee and Manhattan metrics have length two or minimum distance three. This problem and related topics were subject for extensive research in the last forty years. In this paper two product constructions for perfect Lee codes and diameter perfect Lee codes are presented. These constructions yield a large number of nonlinear perfect codes and nonlinear diameter perfect codes in the Lee and Manhattan metrics. A short survey and other related problems on perfect codes in the Lee and the Manhattan metrics are also discussed.
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