Multicovering bounds from supercodes

Let m be a positive integer. The m-covering radius R_m(C) of a code C is the smallest r∈Z such every set of m binary vectors of length n is a subset of at least one ball of radius r around a codeword in C. We derive new lower bounds on R_m(C) for certain C. We consider a fixed linear code C´ and derive lower bounds on the sizes of codes contained in C´ that have given m-covering radius r. This then says that any explicit code C that is contained in C´ and is smaller than the given bound must have m-covering radius greater than r. The sphere bound for multicovering radius is a lower bound on the size of a code with given covering radius. We can generalize it by considering only m-tuples that are contained in a linear supercode of C