3552 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009 Perfect Codes From Cayley Graphs Over Lipschitz Integers Carmen Martínez, Ramón Beivide, and Ernst M. Gabidulin Abstract—The search for perfect error-correcting codes has received intense interest since the seminal work by Hamming. Decades ago, Golomb and Welch studied perfect codes for the Lee metric in multidimensional torus constellations. In this work, we focus our attention on a new class of four-dimensional signal spaces which include tori as subcases. Our constellations are modeled by means of Cayley graphs defined over quotient rings of Lipschitz integers. Previously unexplored perfect codes of length one will be provided in a constructive way by solving a typical problem of vertices domination in graph theory. The codewords of such perfect codes are constituted by the elements of a principal (left) ideal of the considered quotient ring. The generalization of these techniques for higher dimensional spaces is also considered in this work by modeling their signal sets through Cayley–Dickson algebras. Index Terms—Cayley graphs, Lee metric, Lipschitz integers, perfect codes. I. INTRODUCTION D IFFERENT number theory techniques have been widely used for solving applications in coding theory. In par- ticular, certain types of constellations have been modeled by means of quadratic number fields. Gaussian integers have been used to model quadrature amplitude modulation (QAM) con- stellations [8] and Eisenstein–Jacobi integers to model hexag- onal ones [9]. Other techniques from discrete mathematics and graph theory have also been used to model multidimensional constellations. A classical paper by Golomb and Welch [7] de- fines perfect codes by using polyominoes and hypertorus tilings. More recently in [5], flat tori were used with a similar aim. A key concept behind all these models is the definition of a metric over the signal space. Having a suitable metric enables the study of error-correcting codes over these multidimensional constellations. In [7], the well-known Lee metric was employed and in [8], a new metric denoted as the Mannheim distance was introduced by Huber for two-dimensional constellations. Manuscript received January 03, 2008; revised October 22, 2008. Current version published July 15, 2009. The work of C. Martínez and R. Beivide was supported by the Spanish Ministry of Education and Science under Contracts CICYT TIN2004-07440-C02-01, TIN2007-68023-C02-01, CONSOLIDER Project CSD2007-00050 , by the European Network of Excellence on High Per- formance and Embedded Architecture and Compilation, and by the Research Programme of the University of Cantabria under Contract 030.VI16.648. C. Martínez and R. Beivide are with the Department of Electronics and Computers, University of Cantabria, 39005 Santander, Spain (e-mail: carmen.martinez@unican.es; ramon.beivide@unican.es). E. M. Gabidulin is with the Department of Radio Engineering. Moscow Institute of Physics and Technology (State University). 141700 Dolgoprudny, Moscow Region, Russia (e-mail: gab@mail.mipt.ru). Communicated by I. Dumer, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2009.2023733 Although Huber’s work constitutes a relevant contribution, un- fortunately the Mannheim distance is not a true metric as was proved in [13]. In that work, a metric, denoted as the Gaussian distance, was proposed for the definition of multiple perfect error-correcting codes. Moreover, it was proved that Golomb and Welch two-dimensional perfect Lee codes are particular cases of the ones presented in that paper. All these findings were obtained by modeling the constellations by means of Cayley graphs defined over quotients of Gaussian integers. In the present work, we focus our attention on the proposal of perfect codes over higher dimension signal spaces. In particular, we study new four-dimensional constellations looking for per- fect codes. The tools employed in this study are Cayley graphs of degree eight defined over the integer ring of the quaternions, or the Lipschitz integers. Gaussian and Lipschitz integers con- stitute the integer rings of the complex and quaternion numbers, which are particular cases of Cayley–Dickson algebras. Moti- vated by this fact, we also extend our study to higher dimension signal sets by using quotients of Cayley–Dickson algebras and defining new Cayley graphs over them. The main contributions of this work can be summarized as follows. • A new class of perfect error-correcting codes for four-di- mensional constellations is described. To this end, a new metric denoted as the Lipschitz metric is defined as the dis- tance among vertices of certain Cayley graphs defined over the Lipschitz integers. Such graphs are denoted as Lips- chitz graphs. • A new relationship between the Lee and Lipschitz met- rics will be established in two ways. First, it is proved that four-dimensional Lee tori are particular members of our Lipschitz graphs. Hence, the Lee metric is included, as a particular case, in the more general Lipschitz metric. Second, we consider Lipschitz constellations that can be embedded into two-dimensional tori. In this way, we pro- vide the possibility of using different metrics and then, dif- ferent codes, over the same signal space. • It will be shown that higher degree Cayley graphs can be defined over any algebra obtained with the Cayley–Dickson construction. Consequently, new metrics induced by such graphs can be defined over higher dimen- sion constellations. We prove that perfect codes cannot be defined over such signal sets when using the techniques employed in this work for lower dimension constellations. • An original methodology, which combines concepts and methods of number theory and graph theory, has been used in the search for perfect codes over Lipschitz integers. The remainder of this paper is organized as follows. Section II is devoted to previous and related work. In Section III, perfect 0018-9448/$25.00 © 2009 IEEE