## 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 **F**_{2} for *m* ≥ 6 with

*s*= 2^{⌊m/2⌋}+ 1 for*m*mod 4 ≠ 2 and*s*= 2^{m/2}− 2 for*m*mod 4 = 2

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

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

[4, Theorem 8] gives a construction for digital (*m* – 7, *m*, 2^{r} − 1)-nets over ℤ_{2} with *m* = 3*r* + 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*= (3^{m/2}+ 1)/2 for*m*mod 4 = 0, and*s*= (3^{m/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*= (2^{m}+ 1)/3 for*m*odd, 7 ≤*m*≤ 19 [6, Theorem 3.5],*s*= (2^{m}− 1)/3 for*m*even, 6 ≤*m*≤ 16 [6, Theorem 3.1], and*s*= (2^{m−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* = (2^{m} − 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: 2008-04-04.
http://mint.sbg.ac.at/desc_NBCHEmbeddings-nonconstructive.html