Setting

a > - 2a |

Since we have assumed that

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We proceed to the proof of Theorem 2.7.

**Proof of Theorem 2.7. ** Following convention, we let
denote the number of errors that the codes of a
*G*-family can correct.
It is well-known that each vector in
_{2}^{nm-1}
of weight *t* must be a unique coset leader for
*G*_{ m-1}. Moreover, using our assumption about *n*_{0} (which also holds for
*n*_{m}, since
*n*_{m} *n*_{0}), Lemma 2.8 implies
that

The left-hand side of (2.13) is the number of vectors of weight

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

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The result in Theorem 2.7 also applies to all
G-families trivially seeded with the code
_{d}.

If a *G*-family is trivially seeded, then, necessarily, *n*_{0} = *d* and
.Thus,

We now turn our attention to generating mappings which produce codes
whose information rate is locally maximized. More specifically, for
any code
with covering radius
, we will consider only
generating mappings *f* with the property that the Hamming distance
from
to
is exactly
. We will call such
mappings minimal generating mappings, and the corresponding
family of codes minimal G-codes, because they locally
minimize length (and hence locally maximize information rate) for a
given dimension and minimum-distance.

As an example, the generating mappings for the traditional lexicodes
and the trellis-oriented lexicodes are both minimal. We may now
easily strengthen Lemma 2.6 by observing in its proof
that
= *d* - for minimal *G*-codes.

The covering radius bounds we have developed for *G*-codes
translates naturally to length bounds as well.

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