Tower of Function Fields by García and Stichtenoth

Let b = q2 be the square of a prime power q. In [1] García and Stichtenoth consider the tower F1 ⊆ F2 ⊆ ⋯ of global function fields over Fb, where F1 is the rational function field Fb(x1) and Fi+1 := Fi(zi+1) for i = 1, 2,…, where zi+1 satisfies the equation

zi+1q + zi+1 = xiq+1        with        xi = $\displaystyle {\frac{{z_{i}}}{{x_{i−1}}}}$.

Thus, F2 is the Hermitian function field over Fb.

Let gi := g(Fi/Fb) and Ni := N(Fi/Fb). Then it is shown that

gi = qi + qi−1 +1 – \begin{displaymath}\begin{cases}q^{h+1}+2q^{h} & \textrm{if $i=2h+1$\ is odd},… …ac{3}{2}q^{h}+q^{h−1} & \textrm{if $i=2h$\ is even}.\end{cases}\end{displaymath}

If the characteristic is odd, we have

Ni = (q2 – 1)qi−1 + 2q

for i ≥ 3. For even characteristic we get

N3 = (q2 – 1)q2 + 2q,
N4 = (q2 – 1)q3 + q2 + q,

and

Ni = (q2 – 1)qi−1 +2q2

for i ≥ 5.

Optimality

We have $ \lim_{{i\to\infty}}^{}$Ni/gi = 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 Fq2 with

t ≤ $\displaystyle {\frac{{q}}{{q−1}}}$s – $\displaystyle \sqrt{{\frac{q^{2}s}{q^{2}−1}}}$

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 ≤ $\displaystyle {\frac{{3q−1}}{{q}}}$s – $\displaystyle {\frac{{2q\sqrt{s}}}{{\sqrt{q^{2}−1}}}}$

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

See Also

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

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