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The covering problem in the \( n \)-dimensional \( q \)-ary Hamming space consists of the determination of the minimal cardinality \( K_q(n, R) \) of an \( R \)-covering code. It is known that the sphere covering bound can be improved by considering decompositions of the underlying space, leading to integer programming problems. We describe the method in an elementary way and derive about 50 new computational and theoretical records for lower bounds on \( K_q(n, R) \).