### Bounding Length

By summing Theorem 2.7 over many iterations, we may obtain a lower bound on the length nm of the m-th code of any minimal G-family.

Corollary 2..11   For any trivially seeded, minimal G-code,

nm  m + -

Proof.  Consider summing a weaker inequality obtained from Theorem 2.7 by eliminating the floor function. Summing over iterations 1...m of the G-construction we get:

 d--2 m(d - 2) -

Noting that for trivially seeded codes, and 0, we may reduce this to:

m(d - 2) - ( -  m(d - 2) -

Furthermore, since we are dealing with minimal G-codes,

nm  =  d -   =  md -

so that we may now conclude the proof:

nm  =  md -    md - m(d - 2) -    (d + 4) +

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In the case of a G-family seeded by a non-trivial code of length n0 and covering radius , Corollary 2.11 may be easily generalized to:

nm   n0 + +

Clearly, Corollary 2.11 applies to all lexicodes and trellis-oriented lexicodes. It is a asymptotically tighter than the similar bound given by Brualdi and Pless [9, Theorem3.5]:

 k (9)

Specifically, Corollary 2.11 asymptotically binds k 3 whereas Equation (2.14) binds k 2. Nevertheless, both of these bounds are weak as illustrated by Appendix A.1.

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