Bounds for short covering codes and reactive tabu search

Given a prime power q , c q ( n , R ) denotes the minimum cardinality of a subset H in F q n such that every word in this space differs in at most R coordinates from a multiple of a vector in H . In this work, two new classes of short coverings are established. As an application, a new optimal recor...

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Bibliographic Details
Published inDiscrete Applied Mathematics Vol. 158; no. 5; pp. 522 - 533
Main Authors Mendes, Carlos, Monte Carmelo, Emerson L., Poggi, Marcus
Format Journal Article
LanguageEnglish
Published Elsevier B.V 06.03.2010
Subjects
Covering code
Covering radius
Dominating set
Finite field
Tabu search
Finite field
Tabu search
Dominating set
Covering code
Covering radius
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Summary:Given a prime power q , c q ( n , R ) denotes the minimum cardinality of a subset H in F q n such that every word in this space differs in at most R coordinates from a multiple of a vector in H . In this work, two new classes of short coverings are established. As an application, a new optimal record-breaking result on the classical covering code is obtained by using short covering. We also reformulate the numbers c q ( n , R ) in terms of dominating set on graphs. Departing from this reformulation, the reactive tabu search (a variation of tabu search heuristics) is developed to obtain new upper bounds on c q ( n , R ) . The algorithm is described and conclusions on the results are drawn; they identify the advantages of using the reactive mechanism for this problem. Tables of lower and upper bounds on c q ( n , R ) , q = 3 , 4 , n ≤ 7 , and R ≤ 3 , are also presented.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2009.11.006