MinT - Tower of Function Fields by Bezerra, García, and Stichtenoth

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 F_b, where F₁ is the rational function field F_b(x₁) and F_i := F_i−1(x_i) for i = 2, 3,…, where x_i 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 g_i := g(F_i/F_b) and N_i := N(F_i/F_b). Then it is shown in [1, Theorem 2.9] that

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

if i≡0 mod 4,

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

if i≡2 mod 4, and

g_i = $\displaystyle {\frac{{1}}{{2(q−1)}}}$ (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

N_i ≥ qⁱ(q + 1).

Now it is easy to see that

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

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 N_i (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 F_q³ 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: 2015-09-03. http://mint.sbg.ac.at/desc_FBezerraGarciaStichtenothTower.html

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