On the complexity of multicovering radii

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 Σ₂^P-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 n^ε 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 Σ₂^P,then the m-covering radius of a linear code cannot be approximated within a constant factor or within a factor n^ε where n is the length of the code and ε<1, in polynomial time.