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On the complexity of multicovering radii

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1 Author(s)
Mertz, A. ; Univ. of Kentucky, Lexington, KY, USA

The multicovering radius is a generalization of the covering radius. In this correspondence, we show that lower-bounding the m-covering radius of an arbitrary binary code is NP-complete when m is polynomial in the length of the code. Lower-bounding the m-covering radius of a linear code is Σ2P-complete when m is polynomial in the length of the code. If P is not equal to NP, then the m-covering radius of an arbitrary binary code cannot be approximated within a constant factor or within a factor nε where n is the length of the code and ε<1, in polynomial time. Note that the case when m=1 was also previously unknown. If NP is not equal to Σ2P,then the m-covering radius of a linear code cannot be approximated within a constant factor or within a factor nε where n is the length of the code and ε<1, in polynomial time.

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Information Theory, IEEE Transactions on  (Volume:50 ,  Issue: 8 )