On covering problems of codes
 M. Frances,
 A. Litman
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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 NPcomplete.
This result is established as follows. TheRadius of a binary codeC is the smallest integerr such thatC is contained in a radiusr 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=(v _{1} v _{2} …v _{2n })  ∀i:v _{2i }=v _{2i−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 radiusn ball centered atv; moreover,Y can be constructed in time polynomial inn.
 E. R. Berlekamp, R. J. McEliece, and H. C. A. Tilborg. On the Inherent Intractability of Certain Coding Problems.IEEE Trans. Inform. Theory, vol. IT24, pp. 384–386, 1978. CrossRef
 G. D. Cohen, M. G. Karpovsky, H. F. Mattson Jr., and J. R. Schatz. Covering Radius—Survey and Recent Results.IEEE Trans. Inform. Theory, vol. IT31, pp. 328–343, 1985. CrossRef
 M. R. Garey and D. S. Johnson.Computers and Intractability, A Guide to the Theory of NPCompleteness. Freeman, San Francisco, CA, 1979.
 M. Karpovsky. Weight Distribution of Translates, Covering Radius and Perfect Codes ….IEEE Trans. Inform. Theory, vol. IT27, pp. 462–472, 1981. CrossRef
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 Title
 On covering problems of codes
 Journal

Theory of Computing Systems
Volume 30, Issue 2 , pp 113119
 Cover Date
 19970401
 DOI
 10.1007/BF02679443
 Print ISSN
 14324350
 Online ISSN
 14330490
 Publisher
 SpringerVerlag
 Additional Links
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 Authors

 M. Frances ^{(1)}
 A. Litman ^{(1)}
 Author Affiliations

 1. Department of Computer Science, Technion, 32000, Haifa, Israel