## Tower of Function Fields by Niederreiter and Xing Based on the Tower by García and Stichtenoth

Let *b* be a square with *b* = *q*^{2} and *q* = *p*^{r} with *p* prime. Let *F*_{1} ⊂ *F*_{2} ⊂ ⋯ be the tower of function fields by García and Stichtenoth over **F**_{q2} introduced in [1].

In the proof of [2, Theorem 4], Niederreiter and Xing show that this tower can be complemented with intermediate fields *K*_{n,i} for *i* = 0,…, *r* and *n* ≥ 1 such that for all *n* ≥ 1 we have

*F*

_{n}=

*K*

_{n,0}⊂

*K*

_{n,1}⊂ ⋯ ⊂

*K*

_{n,r}=

*F*

_{n+1}

and [*K*_{n,i} : *K*_{n,i−1}] = *p* for all *i* = 1,…, *r*. Since both towers are Galois extensions, the bound

*N*(

*K*

_{n,i}/

**F**

_{q2}) ≥

*N*(

*F*

_{n+1}/

**F**

_{q2})

for *N*(*K*_{n,i}/**F**_{q2}) in terms of *N*(*F*_{n+1}/**F**_{q2}) is easily established. In addition to that, the exact formula

*g*(

*K*

_{n,i}/

**F**

_{q2}) = (

*g*(

*F*

_{n+1}/

**F**

_{q2}) – 1) –

*q*

^{⌊n/2⌋−1}(

*q*+ 2)(

*q*–

*p*

^{i}) + 1

for *g*(*K*_{n,i}/**F**_{q2}) in terms of *g*(*F*_{n+1}/**F**_{q2}) is derived.

### Usage in the Context of Digital Sequences

Using this tower and Niederreiter-Xing sequence construction II/III, [2, Theorem 4] concludes that a digital (*t*, *s*)-sequence over **F**_{q2} exists with

*t*≤

*s*

for all *s* ≥ 1.

### 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] | Chaoping Xing and Harald Niederreiter. A construction of low-discrepancy sequences using global function fields. Acta Arithmetica, 73(1):87–102, 1995.MR1358190 (96g:11096) |

### 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 Niederreiter and Xing Based on the Tower by García and Stichtenoth.”
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Version: 2008-04-04.
http://mint.sbg.ac.at/desc_FGarciaStichtenothNXTower.html