ISSN 0032-9460, Problems of Information Transmission, 2008, Vol. 44, No. 1, pp. 1–11.
Pleiades Publishing, Inc., 2008.
Original Russian Text
C. Mart´ınez, E. Staﬀord, R. Beivide, E.M. Gabidulin, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44,
No. 1, pp. 3–14.
Modeling Hexagonal Constellations
with Eisenstein–Jacobi Graphs
, and E. M. Gabidulin
Universidad de Cantabria, Santander, Spain
Moscow Institute of Physics and Technology (State University)
Received October 18, 2006; in ﬁnal form, November 1, 2007
Abstract—A set of signal points is called a hexagonal constellation if it is possible to deﬁne a
metric so that each point has exactly six neighbors at distance 1 from it. As sets of signal points,
quotient rings of the ring of Eisenstein–Jacobi integers are considered. For each quotient ring,
the corresponding graph is deﬁned. In turn, the distance between two points of a quotient ring
is deﬁned as the corresponding graph distance. Under certain restrictions, a quotient ring is a
hexagonal constellation with respect to this metric. For the considered hexagonal constellations,
some classes of perfect codes are known. Usinggraphsleadstoanewwayofconstructingthese
codes based on solving a standard graph-theoretic problem of ﬁnding a perfect dominating set.
Also, a relation between the proposed metric and the well-known Lee metric is considered.
Problems of designing error-correcting codes for multi-dimensional signal spaces are intensively
studied. For two-dimensional spaces, in  the Mannheim metric (an analog of the well-known
Manhattan metric) was introduced, and error-correcting codes in this metric were proposed. In this
case, sets of signal points are elements of quotient rings of the ring of Gaussian integers. In ,
this approach was extended to the case of hexagonal constellations, which were modeled by means
of quotient rings of Eisenstein–Jacobi integers, and an upgraded Mannheim metric was proposed
for this case.
In , lattices and plane tessellations were employed to derive codes based on the distance
induced by the Euclidean metric over certain graphs embedded in ﬂat tori. Both two-dimensional
and hexagonal constellations were considered. It was pointed out that the metrics deﬁned in [1, 2]
are particular cases of those used to construct the codes.
In , the circulant graph distance has been proved to be a suitable metric for designing codes
over Gaussian integers. In this metric, perfect group codes were constructed. The methodology
described in  can also be applied to deﬁne perfect codes over hexagonal constellations. To this
aim, we introduce a family of degree-six graphs whose vertices are labeled with elements of quotient
rings of the ring of Eisenstein–Jacobi integers. This allows us to interpret the distance between
graph vertices as a distance between elements of a quotient ring and to consider codes over the ring
Supported in part by the Spanish Comision Interministerial de Ciencia y Tecnologia (CICYT), Grant
TIN2004-07440-C02-01, and the Research Program of the Universidad de Cantabria, Grant 30.640.VI07.
Supported in part by the Russian Foundation for Basic Research, project no. GFENa-05-01-39017.