Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension

With Niederreiter-Xing’s sequence construction III a digital (t, s)-sequence over Fb can be constructed based on an algebraic function field of genus g containing s + 1 distinct places P, P1,…, Ps with deg P = 1. The t-parameter is bounded by

tg + $\displaystyle \sum_{{i=1}}^{{s}}$(deg Pi − 1).

In the proof of [1, Theorem 3] Niederreiter and Xing 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+1b + zi+1 = xib+1        with        xi+1 = $\displaystyle {\frac{{z_{i+1}}}{{x_{i}}}}$.

Now let Ei := FiFb2, then E1E2 ⊆ ⋯ is the tower of function fields over Fb2 discovered by García and Stichtenoth in [2], thus exact formulas for g(Ei/Fb2) and N(Ei/Fb2) are known. It follows from the invariance of the genus under constant field extensions that

gi := g(Fi/Fb) = g(Ei/Fb2).

Furthermore

N(Fi/Fb) + 2N2(Fi/Fb) = N(Ei/Fb2),

where N2 denotes the number of places of degree 2, because Ei/Fi is an unramified extension of degree 2. Furthermore it is shown in the proof of [1, Theorem 3] that N(Fi/Fb) ≥ 1, which implies that N(Fi/Fb) ≥ 2 because N(Ei/Fb2) is even, implying that

N(Fi/Fb) + N2(Fi/Fb) ≥ 2 + (N(Ei/Fb2) − 2)/2.

Thus a (gi − 1 + s, s)-sequence over Fb can be constructed for all s = 1,…, 1 + (N(Ei/Fb2) − 2)/2 and for all i ≥ 1.

Related Result

Based on this construction it is shown in [1, Theorem 3] that a digital (t, s)-sequence over Fb with

t$\displaystyle {\frac{{3b−1}}{{b−1}}}$ s$\displaystyle {\frac{{(2b+4)\sqrt{s−1}}}{{\sqrt{b^{2}−1}}}}$ + 2

exists for all prime powers b and all s ≥ 1.

References

[1]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)
[2]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

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. “Niederreiter–Xing Sequence Construction III with García–Stichtenoth Tower As Constant Field Extension.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_SNX3GSTower.html

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