# On covering problems of codes

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## Abstract

Let*C* be a binary code of length*n* (i.e., a subset of {0, 1}^{n}). The*Covering Radius of C* is the smallest integer*r* such that each vector in {0, 1}^{n} is at a distance at most*r* 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. The*Radius* of a binary code*C* is the smallest integer*r* such that*C* 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 of*doubled vectors* of length 2*n*: {*v*=(*v*_{1}*v*_{2} …*v*_{2n}) | ∀*i*:*v*_{2i}=*v*_{2i−1}}. We show that there is a set*Y* ⊂ {0, 1}^{2n} such that for every*v* ε {0, 1}^{2n}:*v* is doubled iff*Y* is contained in the radius-*n* ball centered at*v*; moreover,*Y* can be constructed in time polynomial in*n*.

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