Existence of Nets from Net-Embeddable BCH Codes

Families of digital (mk, m, s)-nets with small k can be constructed by embedding orthogonal arrays obtained from families of BCH-Codes in nets.

Nets over the Binary Field

Constructions for digital (m−4, m, s)-nets over F2 for m ≥ 6 with

are given in [1]. For 12 ≤ m ≤ 270 the second case can be improved to s = 2m/2 − 1 according to [2].

[3] gives a construction for digital (m – 5, m, 2m/2–1 + 1)-nets over 2 for all even m ≥ 4, improving an earlier construction from [2], which obtains only s = 2m/2–1 − 1 instead of s = 2m/2–1 + 1.

[4, Theorem 8] gives a construction for digital (m – 7, m, 2r − 1)-nets over 2 with m = 3r + 2 for all r ≥ 2 with gcd(r, 6) = 1.

Nets over the Ternary Field

Ternary digital (m−4, m, s)-nets can be constructed for all m ≥ 5 with

The construction for odd m is from [1], the ones for even m are from [5].

Nets over the Quaternary Field

Quaternary digital (m−4, m, s)-nets can be constructed with

The existence of quaternary digital (m−4, m, s)-nets for even m ≥ 18 with s = (2m − 1)/3 is also given in [6, Theorem 3.1].

See Also

References

[1]Yves Edel and Jürgen Bierbrauer.
Construction of digital nets from BCH-codes.
In Harald Niederreiter, Peter Hellekalek, Gerhard Larcher, and Peter Zinterhof, editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, volume 127 of Lecture Notes in Statistics, pages 221–231. Springer-Verlag, 1998.
[2]Tor Helleseth, Torleiv Kløve, and Vladimir I. Levenshtein.
Hypercubic 4 and 5-designs from double-error-correcting BCH codes.
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A family of binary (t, m, s)-nets of strength 5.
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doi:10.1007/s006050170047
[6]San Ling and Ferruh Özbudak.
Some constructions of (t, m, s)-nets with improved parameters.
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Article in press.
doi:10.1016/j.ffa.2007.09.007
[7]Andrew T. Clayman, Kenneth Mark Lawrence, Gary L. Mullen, Harald Niederreiter, and Neil J. A. Sloane.
Updated tables of parameters of (t, m, s)-nets.
Journal of Combinatorial Designs, 7(5):381–393, 1999.
doi:10.1002/(SICI)1520-6610(1999)7:5<381::AID-JCD7>3.0.CO;2-S MR1702298 (2000d:05014)
[8]Jürgen Bierbrauer, Yves Edel, and Wolfgang Ch. Schmid.
Coding-theoretic constructions for (t, m, s)-nets and ordered orthogonal arrays.
Journal of Combinatorial Designs, 10(6):403–418, 2002.
doi:10.1002/jcd.10015

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. “Existence of Nets from Net-Embeddable BCH Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_NBCHEmbeddings-nonconstructive.html

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