## Tower of Function Fields by García and Stichtenoth

Let b = q2 be the square of a prime power q. In  García and Stichtenoth consider the tower F1F2 ⊆ ⋯ of global function fields over Fb, where F1 is the rational function field Fb(x1) and Fi+1 := Fi(zi+1) for i = 1, 2,…, where zi+1 satisfies the equation

zi+1q + zi+1 = xiq+1        with        xi = .

Thus, F2 is the Hermitian function field over Fb.

Let gi := g(Fi/Fb) and Ni := N(Fi/Fb). Then it is shown that

gi = qi + qi−1 +1 – If the characteristic is odd, we have

Ni = (q2 – 1)qi−1 + 2q

for i ≥ 3. For even characteristic we get

N3 = (q2 – 1)q2 + 2q,
N4 = (q2 – 1)q3 + q2 + q,

and

Ni = (q2 – 1)qi−1 +2q2

for i ≥ 5.

### Optimality

We have Ni/gi = q−1, thus this tower attains the Drinfelʹd-Vlăduţ bound  and is therefore asymptotically optimal.

### Usage in the Context of Digital Sequences

In [3, Theorem 5] this result and Niederreiter-Xing sequence construction II/III are used for constructing a digital (t, s)-sequence over Fq2 with

t s for all s ≥ 1.

Based thereupon and on base reduction, in [3, Proposition 6] a (t, s)-sequence in base q is constructed with

t s for all s ≥ 1. Digital sequences with the same leading coefficient can be obtained using Niederreiter-Xing sequence construction III.