Tower of Function Fields by Bezerra, García, and Stichtenoth

Let b = q3 be a cube of a prime power q. In [1] Bezerra, 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 := Fi−1(xi) for i = 2, 3,…, where xi satisfies the equation

$\displaystyle {\frac{{1-x_{i}}}{{x_{i}^{q}}}}$ = $\displaystyle {\frac{{x_{i−1}^{q}+x_{i−1}−1}}{{x_{i−1}}}}$.

Let gi := g(Fi/Fb) and Ni := N(Fi/Fb). Then it is shown in [1, Theorem 2.9] that

gi = $\displaystyle {\frac{{1}}{{2(q−1)}}}$(qi+1 +2qi – 2q(i+2)/2 – 2qi/2 + q) – iq(i−2)/2(q + 1)/4

if i≡0 mod 4,

gi = $\displaystyle {\frac{{1}}{{2(q−1)}}}$(qi+1 +2qi – 4q(i+2)/2 + q) – (i – 2)q(i−2)/2(q + 1)/4

if i≡2 mod 4, and

gi = $\displaystyle {\frac{{1}}{{2(q−1)}}}$(qi+1 +2qi – q(i+3)/2 – 3q(i+1)/2 + q) – (i – 1)q(i−2)/2/2

if i≡1 mod 2. Furthermore it is shown in [1, Theorem 3.2] that

Ni ≥ qi(q + 1).

Now it is easy to see that

$\displaystyle \lim_{{i\to\infty}}^{}$Ni/gi ≥ $\displaystyle {\frac{{2(q^{2}−1)}}{{q+2}}}$,

therefore this tower attains the asymptotic rate of Zink’s existence result [2] if q is prime and establishes an equivalent result if q is not prime.

The special case with b = 8 and q = 2 has been studies earlier by van der Geer and van der Vlugt. In this case the actual value of Ni (instead of a lower bound) is known.

Usage in the Context of Digital Sequences

In [3, Theorem 4.6] this result and Niederreiter-Xing sequence construction II/III are used for constructing a digital (t, s)-sequence over Fq3 with

t ≤ $\displaystyle {\frac{{q(q+2)}}{{2(q^{2}−1)}}}$ s

for all s ≥ 1.

References

[1]Juscelino Bezerra, Arnaldo García, and Henning Stichtenoth.
An explicit tower of function fields over cubic finite fields and Zink’s lower bound.
Journal für die reine und angewandte Mathematik, 589:159–199, December 2005.
doi:10.1515/crll.2005.2005.589.159
[2]Thomas Zink.
Degeneration of Shimura surfaces and a problem in coding theory.
In L. Budach, editor, Fundamentals of Computation, volume 199 of Lecture Notes in Computer Science, pages 503–511. Springer, Berlin, 1985.
[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 Bezerra, García, and Stichtenoth.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_FBezerraGarciaStichtenothTower.html

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