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 $\ensuremath{{\mathfrak{G}}}$-family.

Corollary 2   For any trivially seeded, minimal $\ensuremath{{\mathfrak{G}}}$-code,

nm  $\displaystyle \leq$ $\displaystyle \left(\vphantom{m+\frac{1}{2}}\right.$m + $\displaystyle {\textstyle\frac{1}{2}}$$\displaystyle \left.\vphantom{m+\frac{1}{2}}\right)$$\displaystyle {\frac{d+4}{3}}$ - $\displaystyle {\textstyle\frac{2}{3}}$.

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

$\displaystyle \sum_{i=1}^{m}$$\displaystyle \rho_{i}^{}$ $\displaystyle \geq$ $\displaystyle \sum_{i=1}^{m}$$\displaystyle \left(\vphantom{{d - \frac{\rho_{i-1}}{2} - 2}}\right.$d-$\displaystyle {\frac{\rho_{i-1}}{2}}$-2$\displaystyle \left.\vphantom{{d - \frac{\rho_{i-1}}{2} - 2}}\right)$  
  $\displaystyle \geq$ m(d - 2) - $\displaystyle {\textstyle\frac{1}{2}}$$\displaystyle \sum_{i=0}^{{m}-1}$$\displaystyle \rho_{i}^{}$.  

Noting that for trivially seeded codes, $\rho_0 = \ensuremath {\left\lfloor \frac{d}{2} \right\rfloor}$ and $ \rho_{m}^{}$ $ \geq$ 0, we may reduce this to:

$\displaystyle {\textstyle\frac{3}{2}}$$\displaystyle \sum_{i=1}^{{m}}$$\displaystyle \rho_{i}^{}$  $\displaystyle \geq$ I>m(d - 2) - $\displaystyle {\textstyle\frac{1}{2}}$($\displaystyle \rho_{0}^{}$ - $\displaystyle \rho_{m}^{}$$\displaystyle \geq$ I>m(d - 2) - $\displaystyle {\frac{d}{4}}$.

Furthermore, since we are dealing with minimal $\ensuremath{{\mathfrak{G}}}$-codes,

nm  = tex2html_image_mark>#tex2html_wrap_indisplay7503#$\displaystyle \left(\vphantom{ d-\rho_i }\right.$d - $\displaystyle \rho_{i}^{}$$\displaystyle \left.\vphantom{ d-\rho_i }\right)$  = I>md - $\displaystyle \sum_{i=1}^{m}$$\displaystyle \rho_{i}^{}$

so that we may now conclude the proof:

nm = md - $\displaystyle \sum_{i=1}^{m}$$\displaystyle \rho_{i}^{}$    
  $\displaystyle \leq$ md - $\displaystyle {\textstyle\frac{2}{3}}$$\displaystyle \left(\vphantom{ {m}(d-2) - \frac{d}{4}}\right.$m(d - 2) - $\displaystyle {\frac{d}{4}}$$\displaystyle \left.\vphantom{ {m}(d-2) - \frac{d}{4}}\right)$    
  $\displaystyle \leq$ $\displaystyle {\frac{m}{3}}$(d + 4) + $\displaystyle {\frac{d}{6}}$.    

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In the case of a $\ensuremath{{\mathfrak{G}}}$-family seeded by a non-trivial code of length n0 and covering radius $ \rho_{0}^{}$, Corollary 2 may be easily generalized to

nm  $\displaystyle \leq$  n0 + $\displaystyle {\frac{m(d+4)+\rho_0}{3}}$ + $\displaystyle {\frac{d}{6}}$.

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 $\displaystyle \geq$ \begin{displaymath}\begin{cases}
n-2-\ensuremath {\left\lfloor \log_2(n-1) \rig...
...d-4} \right\rfloor}&\text{if } d \equiv 2 \pmod{4}. \end{cases}\end{displaymath} (12)

Specifically, Corollary 2 asymptotically binds k $ \geq$ 3$ {\frac{n}{d}}$ whereas Equation (15) binds k $ \geq$ 2$ {\frac{n}{d}}$. Nevertheless, both of these bounds are weak as illustrated by Appendix .1.