Tower of Function Fields by Niederreiter and Xing Based on the Tower by García and Stichtenoth

Let b be a square with b = q2 and q = pr with p prime. Let F1F2 ⊂ ⋯ be the tower of function fields by García and Stichtenoth over Fq2 introduced in [1].

In the proof of [2, Theorem 4], Niederreiter and Xing show that this tower can be complemented with intermediate fields Kn,i for i = 0,…, r and n ≥ 1 such that for all n ≥ 1 we have

Fn = Kn,0Kn,1 ⊂ ⋯ ⊂ Kn,r = Fn+1

and [Kn,i : Kn,i−1] = p for all i = 1,…, r. Since both towers are Galois extensions, the bound

N(Kn,i/Fq2) ≥ N(Fn+1/Fq2)

for N(Kn,i/Fq2) in terms of N(Fn+1/Fq2) is easily established. In addition to that, the exact formula

g(Kn,i/Fq2) = (g(Fn+1/Fq2) – 1) – q⌊n/2⌋−1(q + 2)(qpi) + 1

for g(Kn,i/Fq2) in terms of g(Fn+1/Fq2) is derived.

Usage in the Context of Digital Sequences

Using this tower and Niederreiter-Xing sequence construction II/III, [2, Theorem 4] concludes that a digital (t, s)-sequence over Fq2 exists with

t s

for all s ≥ 1.

References

 [1] Arnaldo García and Henning Stichtenoth.A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound.Inventiones Mathematicae, 121(1):211–222, December 1995.doi:10.1007/BF01884295 [2] Chaoping Xing and Harald Niederreiter.A construction of low-discrepancy sequences using global function fields.Acta Arithmetica, 73(1):87–102, 1995.MR1358190 (96g:11096)