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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 T₁ ⊆ T₂ ⊆ ⋯ of global function fields over F_b, where T₁ is the rational function field F_b(x₁) and T_i := T_i−1(x_i) for i = 2, 3,…, where x_i satisfies the equation

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

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

g_i := g(T_i/F_b) = (Q_i – 1)(R_i − 1).

Let P_∞ denote the pole of x₁ and P_α for α ∈ F_b the zero of x₁ – α in T₁ = F_b(x₁). Let Ω := {α ∈ F_b : α^q + α = 0} and Ω^* := Ω ∖ {0}. Then it is shown in [1, Lemma 3.3] that P_∞ and P_α with α ∈ Ω^* are completely ramified in T_i/T₁, whereas P_α with α ∈ F_b ∖ Ω splits completely in every extention T_i/T₁. Therefore the number of F_b-rational points of T_i is

N_i := N(T_i/F_b) = (q² – q)qⁱ⁻¹ +1 + (q – 1) + s_i,

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

Weierstrass Semigroup

The Weierstrass semigroup H_i of the unique place over P_∞ in T_i is H₁ = ℕ₀ for i = 1 and

H_i = qH_i−1∪{ℕ₀ + R_i(Q_i – 1)}

for i ≥ 2 [3].

Optimality

We have $\lim_{{i\to\infty}}^{}$ N_i/g_i = 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: 2015-09-03. http://mint.sbg.ac.at/desc_FGarciaStichtenothTower2.html

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