Modeling hexagonal constellations with Eisenstein-Jacobi graphs

C Martinez, E Stafford, R Beivide… - Problems of Information …, 2008 - Springer
Problems of Information Transmission, 2008Springer
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 …
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. Using graphs leads to a new way of constructing these 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.
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