## Tower of Function Fields by García and Stichtenoth

Let *b* = *q*^{2} be the square of a prime power *q*. In [1] García and Stichtenoth 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+1} := *F*_{i}(*z*_{i+1}) for *i* = 1, 2,…, where *z*_{i+1} satisfies the equation

*z*

_{i+1}

^{q}+

*z*

_{i+1}=

*x*

_{i}

^{q+1}with

*x*

_{i}= .

Thus, *F*_{2} is the Hermitian function field over **F**_{b}.

Let *g*_{i} := *g*(*F*_{i}/**F**_{b}) and *N*_{i} := *N*(*F*_{i}/**F**_{b}). Then it is shown that

*g*

_{i}=

*q*

^{i}+

*q*

^{i−1}+1 –

If the characteristic is odd, we have

*N*

_{i}= (

*q*

^{2}– 1)

*q*

^{i−1}+ 2

*q*

for *i* ≥ 3. For even characteristic we get

*N*

_{3}= (

*q*

^{2}– 1)

*q*

^{2}+ 2

*q*,

*N*

_{4}= (

*q*

^{2}– 1)

*q*

^{3}+

*q*

^{2}+

*q*,

and

*N*

_{i}= (

*q*

^{2}– 1)

*q*

^{i−1}+2

*q*

^{2}

for *i* ≥ 5.

### Optimality

We have *N*_{i}/*g*_{i} = *q*−1, thus this tower attains the Drinfelʹd-Vlăduţ bound [2] and is therefore asymptotically optimal.

### Usage in the Context of Digital Sequences

In [3, Theorem 5] this result and Niederreiter-Xing sequence construction II/III are used for constructing a digital (*t*, *s*)-sequence over **F**_{q2} with

*t*≤

*s*–

for all *s* ≥ 1.

Based thereupon and on base reduction, in [3, Proposition 6] a (*t*, *s*)-sequence in base *q* is constructed with

*t*≤

*s*–

for all *s* ≥ 1. Digital sequences with the same leading coefficient can be obtained using Niederreiter-Xing sequence construction III.

### See Also

If

*q*is not prime this tower can be refined by including additional intermediate fields.

### References

[1] | Arnaldo García and Henning Stichtenoth. A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound. Inventiones Mathematicae, 121(1):211–222, December 1995.doi:10.1007/BF01884295 |

[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] | Harald Niederreiter and Chaoping Xing. Low-discrepancy sequences and global function fields with many rational places. Finite Fields and Their Applications, 2(3):241–273, July 1996.doi:10.1006/ffta.1996.0016 MR1398076 (97h:11080) |

### 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 and Stichtenoth.”
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Version: 2008-04-04.
http://mint.sbg.ac.at/desc_FGarciaStichtenothTower.html