Technical Report CS0827

TR#:CS0827
Class:CS
Title: ON COVERING PROBLEMS OF CODES.
Authors: M. Frances and A. Litman
PDFCS0827.pdf
Abstract: Let C be a binary code of length n. The Radius of C is the smallest integer r such that C is contained in an r-radius ball in the Hamming metric space <\{0,1\}^n,d>. 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. We show that the problems of computing the Radius and the Covering Radius of an arbitrary binary code are both NP complete. A central tool in our work is an intriguing characterization of the following set of binary vectors of length 2n: \{v=v_1,v_2 \cdots v_{2n}|v_{2i} = v_{2i-1} \forall i=1,...,n\} (doubled vectors). We show that there is a specific set Y of O(n) vectors such that the doubled vectors are exactly the centers of all n-radius spheres which contains Y.
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