A Generalization of Lee Codes

TL;DR

This paper introduces the concept of perfect distance-dominating sets (PDDS) as a generalization of Lee codes and other coding structures, exploring their properties, constructions, non-existence results, and connections to the Golomb-Welch conjecture.

Contribution

It systematically studies PDDSs, constructs examples related to applications, proves non-existence of some types, and extends the Golomb-Welch conjecture within this new framework.

Findings

Constructed lattice-like PDDSs relevant for practical decoding

Proved non-existence of certain PDDS configurations

Extended the Golomb-Welch conjecture using PDDS concepts

Abstract

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.