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