Tower of Function Fields over F8 by van der Geer and van der Vlugt

In [1] van der Geer and van der Vlugt consider the tower F1 ⊆ F2 ⊆ ⋯ of global function fields over F8, where F1 is the rational function field F8(x1) and Fi := Fi−1(xi) for i = 2, 3,…, where xi satisfies the equation

xi2 + xi = xi−1 +1 + $\displaystyle {\frac{{1}}{{x_{i−1}}}}$.

Let gi := g(Fi/F8) and Ni := N(Fi/F8). Then it is shown in [1, Theorem 4.2] that

gi = 2i+1 +1 – 2⌊(i−1)/2⌋−1⋅\begin{displaymath}\begin{cases}i+2\lfloor(i−1)/4\rfloor+14 & \textrm{if $i$\ is even},\\ i+9 & \textrm{if $i$\ is odd}.\end{cases}\end{displaymath}

and in [1, Theorem 4.3] that Ni = 3⋅2i + 2.

Now it is easy to see that

$\displaystyle \lim_{{i\to\infty}}^{}$Ni/gi = 3/2,

therefore this tower has the asymptotic rate of Zink’s existence result [2].

This tower is in fact the special case for b = 8 of the tower by Bezerra, García, and Stichtenoth.

References

[1]Gerard van der Geer and Marcel van der Vlugt.
An asymptotically good tower of curves over the field with eight elements.
The Bulletin of the London Mathematical Society, 34(3):291–300, 2002.
doi:10.1112/S0024609302001017
[2]Thomas Zink.
Degeneration of Shimura surfaces and a problem in coding theory.
In L. Budach, editor, Fundamentals of Computation, volume 199 of Lecture Notes in Computer Science, pages 503–511. Springer, Berlin, 1985.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Tower of Function Fields over F8 by van der Geer and van der Vlugt.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_FGeerVlugtTower.html

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