Second 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 T1T2 ⊆ ⋯ of global function fields over Fb, where T1 is the rational function field Fb(x1) and Ti := Ti−1(xi) for i = 2, 3,…, where xi satisfies the equation

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

Let Qi := q⌊i/2⌋ and Ri := q⌈i/2⌉. Then it is shown in [1, Remark 3.8] that

gi := g(Ti/Fb) = (Qi – 1)(Ri − 1).

Let P denote the pole of x1 and Pα for αFb the zero of x1α in T1 = Fb(x1). Let Ω := {αFb  :  αq + α = 0} and Ω* := Ω ∖ {0}. Then it is shown in [1, Lemma 3.3] that P and Pα with α ∈ Ω* are completely ramified in Ti/T1, whereas Pα with αFb ∖ Ω splits completely in every extention Ti/T1. Therefore the number of Fb-rational points of Ti is

Ni := N(Ti/Fb) = (q2q)qi−1 +1 + (q – 1) + si,

where si denotes the number of rational places over P0 in Fi. Obviously, s1 = 1. For even q ≥ 4 the exact value of si is given in [2] as s2 = q, s3 = q2, and si = 2q2q for i ≥ 4.

Weierstrass Semigroup

The Weierstrass semigroup Hi of the unique place over P in Ti is H1 = ℕ0 for i = 1 and

Hi = qHi−1∪{ℕ0 + Ri(Qi – 1)}

for i ≥ 2 [3].

Optimality

We have $ \lim_{{i\to\infty}}^{}$Ni/gi = q−1 [1, Theorem 3.1], thus this tower attains the Drinfelʹd-Vlăduţ bound [4] and is therefore asymptotically optimal.

References

[1]Arnaldo García and Henning Stichtenoth.
On the asymptotic behaviour of some towers of function fields over finite fields.
Journal of Number Theory, 61(2):248–273, December 1996.
doi:10.1006/jnth.1996.0147
[2]Ilia Aleshnikov, P. Vijay Kumar, Kenneth W. Shum, and Henning Stichtenoth.
On the splitting of places in a tower of function fields meeting the Drinfeld-Vlăduţ bound.
IEEE Transactions on Information Theory, 47(4):1613–1619, May 2001.
doi:10.1109/18.923746 MR1830111 (2002c:94042)
[3]Ruud Pellikaan, Henning Stichtenoth, and Fernando Torres.
Weierstrass semigroups in an asymptotically good tower of function fields.
Finite Fields and Their Applications, 4(4):381–392, October 1998.
doi:10.1006/ffta.1998.0217 MR1648573 (99g:11139)
[4]Sergei G. Vlăduţ and Vladimir Gershonovich Drinfelʹd.
Number of points of an algebraic curve.
Functional Analysis and its Applications, 17:53–54, 1983.

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. “Second Tower of Function Fields by García and Stichtenoth.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_FGarciaStichtenothTower2.html

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