## 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 F1F2 ⊆ ⋯ 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

= .

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

gi = (qi+1 +2qi – 2q(i+2)/2 – 2qi/2 + q) – iq(i−2)/2(q + 1)/4

if i≡0 mod 4,

gi = (qi+1 +2qi – 4q(i+2)/2 + q) – (i – 2)q(i−2)/2(q + 1)/4

if i≡2 mod 4, and

gi = (qi+1 +2qiq(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

Niqi(q + 1).

Now it is easy to see that

Ni/gi,

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 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