## Tower of Function Fields by García, Stichtenoth, and Rück

Let *b* = *p*^{2} be a square of an odd prime *p*. In [1] García, Stichtenoth, and Rück consider the tower *F*_{1} ⊆ *F*_{2} ⊆ ⋯ of global function fields over **F**_{b}, where *F*_{1} is the rational function field **F**_{b}(*x*_{1}) and *F*_{i} := *F*_{i−1}(*x*_{i}) for *i* = 2, 3,…, where *x*_{i} satisfies the equation

*x*

_{i}

^{2}= .

Then it is shown that

*g*

_{i}:=

*g*(

*F*

_{i}/

**F**

_{b}) ≤ 2

^{i}+ 1

and that

*N*

_{i}:=

*N*(

*F*

_{i}/

**F**

_{b}) ≥ 2

^{i}(

*p*−1)

for all *i* ≥ 1. Note that only an upper bound on the genus *g*_{i} is obtained.

### Optimality

It is easy to see that

*N*

_{i}/

*g*

_{i}=

*p*−1,

therefore this tower attains the Drinfelʹd-Vlăduţ bound [2].

### Usage in the Context of Digital Sequences

In [3, Theorem 4.1] this result and Niederreiter-Xing sequence construction II/III are used for constructing a digital (*t*, *s*)-sequence over **F**_{p2} with

*t*≤

*s*+ 1

for all *s* ≥ 1.

### References

[1] | Arnaldo García, Henning Stichtenoth, and Hans-Georg Rück. On tame towers over finite fields. Journal für die reine und angewandte Mathematik, 557:53–80, April 2003.doi:10.1515/crll.2003.034 |

[2] | Sergei G. Vlăduţ and Vladimir Gershonovich Drinfelʹd. Number of points of an algebraic curve. Functional Analysis and its Applications, 17:53–54, 1983. |

[3] | David J. S. Mayor and Harald Niederreiter. A new construction of ( t, s)-sequences and some improved bounds on their quality parameter.Acta Arithmetica, 128(2):177–191, 2007.MR2314003 |

### 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 by García, Stichtenoth, and Rück.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_FGarciaStichtenothRueckTower.html