### Bounding Length

By summing Theorem 4 over many iterations, we may obtain a lower bound on the length nm of the m-th code of any minimal -family.

Corollary 2   For any trivially seeded, minimal -code,

nm  m + - .

Proof: Consider summing a weaker inequality obtained from Theorem 4 by eliminating the floor function. Summing over iterations 1...m of the -construction we get:

 d--2 m(d - 2) - .

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

I>m(d - 2) - ( - I>m(d - 2) - .

Furthermore, since we are dealing with minimal -codes,

nm  = tex2html_image_mark>#tex2html_wrap_indisplay7503#d -   = I>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 -family seeded by a non-trivial code of length n0 and covering radius , Corollary 2 may be easily generalized to

nm   n0 + + .

Clearly, Corollary 2 applies to all lexicodes and trellis-oriented lexicodes. It is a asymptotically tighter than the similar bound given by Brualdi and Pless [3, Theorem3.5]:

 k (12)

Specifically, Corollary 2 asymptotically binds k 3 whereas Equation (15) binds k 2. Nevertheless, both of these bounds are weak as illustrated by Appendix .1.