## Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension

With Niederreiter-Xing’s sequence construction III a digital (*t*, *s*)-sequence over **F**_{b} can be constructed based on an algebraic function field of genus *g* containing *s* + 1 distinct places *P*_{∞}, *P*_{1},…, *P*_{s} with deg *P*_{∞} = 1. The *t*-parameter is bounded by

*t*≤

*g*+ (deg

*P*

_{i}− 1).

In the proof of [1, Theorem 3] Niederreiter and Xing 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}

^{b}+

*z*

_{i+1}=

*x*

_{i}

^{b+1}with

*x*

_{i+1}= .

Now let *E*_{i} := *F*_{i}**F**_{b2}, then *E*_{1} ⊆ *E*_{2} ⊆ ⋯ is the tower of function fields over **F**_{b2} discovered by García and Stichtenoth in [2], thus exact formulas for *g*(*E*_{i}/**F**_{b2}) and *N*(*E*_{i}/**F**_{b2}) are known. It follows from the invariance of the genus under constant field extensions that

*g*

_{i}:=

*g*(

*F*

_{i}/

**F**

_{b}) =

*g*(

*E*

_{i}/

**F**

_{b2}).

Furthermore

*N*(

*F*

_{i}/

**F**

_{b}) + 2

*N*

_{2}(

*F*

_{i}/

**F**

_{b}) =

*N*(

*E*

_{i}/

**F**

_{b2}),

where *N*_{2} denotes the number of places of degree 2, because *E*_{i}/*F*_{i} is an unramified extension of degree 2. Furthermore it is shown in the proof of [1, Theorem 3] that *N*(*F*_{i}/**F**_{b}) ≥ 1, which implies that *N*(*F*_{i}/**F**_{b}) ≥ 2 because *N*(*E*_{i}/**F**_{b2}) is even, implying that

*N*(

*F*

_{i}/

**F**

_{b}) +

*N*

_{2}(

*F*

_{i}/

**F**

_{b}) ≥ 2 + (

*N*(

*E*

_{i}/

**F**

_{b2}) − 2)/2.

Thus a (*g*_{i} − 1 + *s*, *s*)-sequence over **F**_{b} can be constructed for all *s* = 1,…, 1 + (*N*(*E*_{i}/**F**_{b2}) − 2)/2 and for all *i* ≥ 1.

### Related Result

Based on this construction it is shown in [1, Theorem 3] that a digital (*t*, *s*)-sequence over **F**_{b} with

*t*≤

*s*– + 2

exists for all prime powers *b* and all *s* ≥ 1.

### References

[1] | 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) |

[2] | 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 |

### 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. “Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension.”
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