A1 Refereed original research article in a scientific journal
The number of completely different optimal identifying codes in the infinite square grid
Authors: Mikko Pelto
Publisher: ELSEVIER SCIENCE BV
Publication year: 2017
Journal: Discrete Applied Mathematics
Journal name in source: DISCRETE APPLIED MATHEMATICS
Journal acronym: DISCRETE APPL MATH
Volume: 233
Issue: 31
First page : 143
Last page: 158
Number of pages: 16
ISSN: 0166-218X
eISSN: 1872-6771
DOI: https://doi.org/10.1016/j.dam.2017.07.012
Abstract
Let G be a graph with vertex set V and edge set E. We call any subset C subset of V an identifying code if the setsI(v) = {c is an element of C | {c, v} is an element of E or c = v}are distinct and non-empty for all vertices v is an element of V. We study identifying codes in the infinite square grid. The vertex set of this graph is Z(2) and two vertices are connected by an edge if the Euclidean distance between these vertices is one. Ben-Haim & Litsyn have proved that the minimum density of identifying code in the infinite square grid is 7/20. Such codes are called optimal. We study the number of completely different optimal identifying codes in the infinite square grid. Two codes are called completely different if there exists an integer n such that non x n-square of one code is equivalent with any n x n-square of the other code. In particular, we show that there are exactly two completely different optimal periodic codes and no optimal identifying code is completely different with both of these two periodic codes. (C) 2017 Elsevier B.V. All rights reserved.
Let G be a graph with vertex set V and edge set E. We call any subset C subset of V an identifying code if the setsI(v) = {c is an element of C | {c, v} is an element of E or c = v}are distinct and non-empty for all vertices v is an element of V. We study identifying codes in the infinite square grid. The vertex set of this graph is Z(2) and two vertices are connected by an edge if the Euclidean distance between these vertices is one. Ben-Haim & Litsyn have proved that the minimum density of identifying code in the infinite square grid is 7/20. Such codes are called optimal. We study the number of completely different optimal identifying codes in the infinite square grid. Two codes are called completely different if there exists an integer n such that non x n-square of one code is equivalent with any n x n-square of the other code. In particular, we show that there are exactly two completely different optimal periodic codes and no optimal identifying code is completely different with both of these two periodic codes. (C) 2017 Elsevier B.V. All rights reserved.
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