, Volume 30, Issue 2, pp 113-119
Date: 30 Jul 2007

On covering problems of codes

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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=(v 1 v 2v 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 radius-n ball centered atv; moreover,Y can be constructed in time polynomial inn.