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

Let b = q3 be a cube of a prime power q. In  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  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

  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  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.  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