Our first bound is a recursive upper bound on the covering radius of the m-th code of an arbitrary G-family. We shall use the term to refer to the corresponding term in Definition 1 for this m-th code.

Lemma 2..6   For any G-family of codes and m > 1,

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 )

for all a2. Since the G-construction implicitly demands that d, we may conclude that any coset leader of G m must have weight no greater than

which bounds the covering radius of G m and proves the lemma.

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

Theorem 2..7   For any G-family of codes whose seed code has length

it necessarily follows that

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

Lemma 2..8   If a > 3b > 0 then

 a > a (9)

Subsections
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