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
|minwt(a | l ), wt | (l )|
which bounds the covering radius of G m and proves the lemma.
With some combinatorial analysis, we can also establish a recursive lower bound on the covering radius of G-codes.
it necessarily follows that
Part of the proof of this theorem rests on a simple combinatorial lemma, which we provide without proof.