By summing Theorem 4 over many iterations, we may
obtain a lower bound on the length n_{m} of the mth code
of any minimal
family.
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:
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,
so that we may now conclude the proof:
n_{m} 
= md  


md  m(d  2)  


(d + 4) + . 

width4pt depth2pt height6pt
In the case of a
family seeded by a nontrivial
code of length n_{0} and covering radius ,
Corollary 2 may be easily generalized to
Clearly, Corollary 2 applies to all lexicodes and
trellisoriented 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.