A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
The number of completely different optimal identifying codes in the infinite square grid
Tekijät: Mikko Pelto
Kustantaja: ELSEVIER SCIENCE BV
Julkaisuvuosi: 2017
Journal: Discrete Applied Mathematics
Tietokannassa oleva lehden nimi: DISCRETE APPLIED MATHEMATICS
Lehden akronyymi: DISCRETE APPL MATH
Vuosikerta: 233
Numero: 31
Aloitussivu: 143
Lopetussivu: 158
Sivujen määrä: 16
ISSN: 0166-218X
eISSN: 1872-6771
DOI: https://doi.org/10.1016/j.dam.2017.07.012
Tiivistelmä
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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