The covering number f_m(n,k) is determined exactly for k=1 and 2, with bounds for k>=3, via a sharp extension of the Combinatorial Nullstellensatz to mB^n and a hyperplane construction using Lagrange inversion.
Alon, Combinatorial Nullstellensatz, Combin
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Covering Hypercube $mB^n$
The covering number f_m(n,k) is determined exactly for k=1 and 2, with bounds for k>=3, via a sharp extension of the Combinatorial Nullstellensatz to mB^n and a hyperplane construction using Lagrange inversion.