**Proof. **Consider constructing
*G*_{ m} from
*G*_{ m-1} using
Theorem 2.5. Any two coset leaders *l* and
(*l* ) of
*G*_{ m-1} must each have weight at most
, by the definition of the covering radius. Consider the
weight of a corresponding coset leader of
*G*_{ m}, based on
Theorem 2.5:

minwt(a | l ), wt | (l ) |
|||

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With some combinatorial analysis, we can also establish a recursive
*lower* bound on the covering radius of *G*-codes.

Part of the proof of this theorem rests on a simple combinatorial lemma, which we provide without proof.