On covering problems of codes

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 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 radius-n ball centered atv; moreover,Y can be constructed in time polynomial inn.