Existence of Nets from Net-Embeddable BCH Codes
Families of digital (m−k, 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
s = 2⌊m/2⌋ + 1 for m mod 4 ≠ 2 and
s = 2m/2 − 2 for m mod 4 = 2
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 ℤ2 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 ℤ2 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
s = 3(m−1)/2 − 1 for m odd,
s = (3m/2 + 1)/2 for m mod 4 = 0, and
s = (3m/2 − 1)/2 for m mod 4 = 2.
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
s = (2m + 1)/3 for m odd, 7 ≤ m ≤ 19 [6, Theorem 3.5],
s = (2m − 1)/3 for m even, 6 ≤ m ≤ 16 [6, Theorem 3.1], and
s = (2m−1 + 1)/3 for even m ≥ 18 [6, Theorem 3.7].
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
The constructions from [1] are listed as Construction 25 in [7].
The constructions from [1] and [5] are listed as Construction “f” in [8].
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. Designs, Codes and Cryptography, 28(3):265–282, 2003. doi:10.1023/A:1024110021836 MR1976961 (2004e:05045) |
| [3] | Jürgen Bierbrauer and Yves Edel. A family of binary (t, m, s)-nets of strength 5. Designs, Codes and Cryptography, 37(2):211–214, November 2005. doi:10.1007/s10623-004-3986-0 |
| [4] | Jürgen Bierbrauer and Yves Edel. Families of nets of low and medium strength. Integers. Electronic Journal of Combinatorial Number Theory, 5(3):#A03, 13 pages, 2005. (electronic). |
| [5] | Yves Edel and Jürgen Bierbrauer. Families of ternary (t, m, s)-nets related to BCH-codes. Monatshefte für Mathematik, 132(2):99–103, May 2001. doi:10.1007/s006050170047 |
| [6] | San Ling and Ferruh Özbudak. Some constructions of (t, m, s)-nets with improved parameters. Finite Fields and Their Applications, 2008. 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: 2015-09-03.
http://mint.sbg.ac.at/desc_NBCHEmbeddings-nonconstructive.html