Theory of Computing Systems

, Volume 30, Issue 2, pp 113–119

On covering problems of codes

Authors

  • M. Frances
    • Department of Computer ScienceTechnion
  • A. Litman
    • Department of Computer ScienceTechnion
Article

DOI: 10.1007/BF02679443

Cite this article as:
Frances, M. & Litman, A. Theory of Computing Systems (1997) 30: 113. doi:10.1007/BF02679443

Abstract

LetC be a binary code of lengthn (i.e., a subset of {0, 1}n). TheCovering Radius of C is the smallest integerr such that each vector in {0, 1}n is at a distance at mostr from some code word. Our main result is that the decision problem associated with the Covering Radius of arbitrary binary codes is NP-complete.

This result is established as follows. TheRadius of a binary codeC is the smallest integerr such thatC is contained in a radius-r ball of the Hamming metric space 〈{0, 1}n,d〉. It is known [K] that the problems of computing the Radius and the Covering Radius are equivalent. We show that the 3SAT problem is polynomially reducible to the Radius decision problem.

A central tool in our reduction is a metrical characterization of the set ofdoubled vectors of length 2n: {v=(v1v2v2n) | ∀i:v2i=v2i−1}. We show that there is a setY ⊂ {0, 1}2n such that for everyv ε {0, 1}2n:v is doubled iffY is contained in the radius-n ball centered atv; moreover,Y can be constructed in time polynomial inn.

Copyright information

© Springer-Verlag New York Inc 1997