Bounds for short covering codes and reactive tabu search

  • Authors:

  • Carlos Mendes;Emerson L. Monte Carmelo;Marcus Poggi

  • Affiliations:

  • Departamento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, Brazil;Departamento de Matemática, Universidade Estadual de Maringá, Brazil;Departamento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, Brazil

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

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.