## 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

• s = 2⌊m/2⌋ + 1 for m mod 4 ≠ 2 and

• s = 2m/2 − 2 for m mod 4 = 2

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

 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 , 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

• 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 , the ones for even m are from .

### 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].

• The constructions from  are listed as Construction 25 in .

• The constructions from  and  are listed as Construction “f” in .

### References

  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.  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)  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  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).  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  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  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)  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