## Tower of Function Fields over **F**_{8} by van der Geer and van der Vlugt

In [1] van der Geer and van der Vlugt consider the tower *F*_{1} ⊆ *F*_{2} ⊆ ⋯ of global function fields over **F**_{8}, where *F*_{1} is the rational function field **F**_{8}(*x*_{1}) and *F*_{i} := *F*_{i−1}(*x*_{i}) for *i* = 2, 3,…, where *x*_{i} satisfies the equation

*x*

_{i}

^{2}+

*x*

_{i}=

*x*

_{i−1}+1 + .

Let *g*_{i} := *g*(*F*_{i}/**F**_{8}) and *N*_{i} := *N*(*F*_{i}/**F**_{8}). Then it is shown in [1, Theorem 4.2] that

*g*

_{i}= 2

^{i+1}+1 – 2

^{⌊(i−1)/2⌋−1}⋅

and in [1, Theorem 4.3] that *N*_{i} = 3⋅2^{i} + 2.

Now it is easy to see that

*N*

_{i}/

*g*

_{i}= 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 **F**_{8} by van der Geer and van der Vlugt.”
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Version: 2008-04-04.
http://mint.sbg.ac.at/desc_FGeerVlugtTower.html