Tower of Function Fields by García, Stichtenoth, and Rück

Let b = p2 be a square of an odd prime p. In [1] García, Stichtenoth, and Rück consider the tower F1 ⊆ F2 ⊆ ⋯ of global function fields over Fb, where F1 is the rational function field Fb(x1) and Fi := Fi−1(xi) for i = 2, 3,…, where xi satisfies the equation

xi2 = $\displaystyle {\frac{{x_{i−1}^{2}+1}}{{2x_{i−1}}}}$.

Then it is shown that

gi := g(Fi/Fb) ≤ 2i + 1

and that

Ni := N(Fi/Fb) ≥ 2i(p−1)

for all i ≥ 1. Note that only an upper bound on the genus gi is obtained.

Optimality

It is easy to see that

$\displaystyle \lim_{{i\to\infty}}^{}$Ni/gi = p−1,

therefore this tower attains the Drinfelʹd-Vlăduţ bound [2].

Usage in the Context of Digital Sequences

In [3, Theorem 4.1] this result and Niederreiter-Xing sequence construction II/III are used for constructing a digital (t, s)-sequence over Fp2 with

t ≤ $\displaystyle {\frac{{2}}{{p−1}}}$ s + 1

for all s ≥ 1.

References

[1]Arnaldo García, Henning Stichtenoth, and Hans-Georg Rück.
On tame towers over finite fields.
Journal für die reine und angewandte Mathematik, 557:53–80, April 2003.
doi:10.1515/crll.2003.034
[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]David J. S. Mayor and Harald Niederreiter.
A new construction of (t, s)-sequences and some improved bounds on their quality parameter.
Acta Arithmetica, 128(2):177–191, 2007.
MR2314003

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, Stichtenoth, and Rück.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_FGarciaStichtenothRueckTower.html

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