Quotients of Gaussian graphs and their application to perfect codes

Authors:
C. Martínez;R. Beivide;C. Camarero;E. Stafford;E. M. Gabidulin
Affiliations:
Electronics and Computers Department, University of Cantabria, Faculty of Sciences, Avda. Los Castros s/n, 39006 Santander, Spain;Electronics and Computers Department, University of Cantabria, Faculty of Sciences, Avda. Los Castros s/n, 39006 Santander, Spain;Electronics and Computers Department, University of Cantabria, Faculty of Sciences, Avda. Los Castros s/n, 39006 Santander, Spain;Electronics and Computers Department, University of Cantabria, Faculty of Sciences, Avda. Los Castros s/n, 39006 Santander, Spain;Department of Radio Engineering, Moscow Institute of Physics and Technology (State University), Institutskii Pereulok, 9 141700 Dolgoprudny, Moscow Region, Russia
Venue:
Journal of Symbolic Computation
Year:
2010

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Abstract

A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes. Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs. Codewords of perfect codes will be represented by vertices of a quotient graph of the Gaussian graph in which the signal space has been defined. It will be shown that any quotient graph of a Gaussian graph is indeed a Gaussian graph. This makes it possible to apply previously known properties of Gaussian graphs to the analysis of perfect codes. To illustrate the modeling power of this graph-based tool, perfect Lee codes will be analyzed in terms of Gaussian graphs and their quotients.