## Tower of Function Fields by Bezerra, García, and Stichtenoth

Let *b* = *q*^{3} be a cube of a prime power *q*. In [1] Bezerra, García, and Stichtenoth consider the tower *F*_{1} ⊆ *F*_{2} ⊆ ⋯ of global function fields over **F**_{b}, where *F*_{1} is the rational function field **F**_{b}(*x*_{1}) and *F*_{i} := *F*_{i−1}(*x*_{i}) for *i* = 2, 3,…, where *x*_{i} satisfies the equation

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}= (

*q*

^{i+1}+2

*q*

^{i}– 2

*q*

^{(i+2)/2}– 2

*q*

^{i/2}+

*q*) –

*iq*

^{(i−2)/2}(

*q*+ 1)/4

if *i*≡0 mod 4,

*g*

_{i}= (

*q*

^{i+1}+2

*q*

^{i}– 4

*q*

^{(i+2)/2}+

*q*) – (

*i*– 2)

*q*

^{(i−2)/2}(

*q*+ 1)/4

if *i*≡2 mod 4, and

*g*

_{i}= (

*q*

^{i+1}+2

*q*

^{i}–

*q*

^{(i+3)/2}– 3

*q*

^{(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}≥

*q*

^{i}(

*q*+ 1).

Now it is easy to see that

*N*

_{i}/

*g*

_{i}≥ ,

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*≤

*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.”
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Version: 2015-09-03.
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