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The multicovering radii of a code are natural generalizations of the covering radius in which the goal is to cover all m-tuples of vectors for some m as cheaply as possible. In this correspondence, we describe several techniques for obtaining lower bounds on the sizes of codes achieving a given multicovering radius. Our main method is a generalization of the method of linear inequalities based on refined weight distributions of the code. We also obtain a linear upper bound on the 2-covering radius. We further study bounds on the sizes of codes with a given multicovering radius that are subcodes of a fixed code. We find, for example, constraints on parity checks for codes with small ordinary covering radius.