*k* = 2 Construction

For all *t* and *b*, a construction of a (*m*−2, *m*, *s*)-net in base *b* with

*s*=

is given in [1]. This result is based on the equivalence between (*m*−2, *m*, *s*)-nets in base *b* and sets of *s* mutually orthogonal *m*-dimensional hypercubes of order *b* [1, Theorem 2]. Even though this method is constructive, it does not necessarily result in a digital net.

In [2, Proposition 2], Niederreiter describes an easy method for constructing a digital (*m*−2, *m*, *s*)-net over **F**_{b} with

*s*= .

The *s* 2×*m* matrices

**C**_{i}=

for *i* = 1,…, *s* are constructed by choosing *s* pairwise linearly independent vectors for **c**_{1}^{(i)} (there are exactly *s* such vectors, e.g., all non-zero vectors with the leftmost non-zero coefficient equal to one) and setting **c**_{2}^{(i)} to any vector linearly independent to **c**_{1}^{(i)} for *i* = 1,…, *s*.

The case where *t* = 0 was already given in [3, Theorem 5.4]. This construction can also be seen as a net-embedding of the dual orthogonal array of a linear [*s*, *s*−*m*, 3]-Hamming code over **F**_{b}, which is first embedded in an ordered orthogonal array with depth 2 using adding a *k*th column and then into a net using net from OOA [4, Section 7].

### See Also

Construction 12 in [5].

### References

[1] | Gary L. Mullen and G. Whittle. Point sets with uniformity properties and orthogonal hypercubes. Monatshefte für Mathematik, 113(4):265–273, December 1992.doi:10.1007/BF01301071 |

[2] | Harald Niederreiter. A combinatorial problem for vector spaces over finite fields. Discrete Mathematics, 96(3):221–228, December 1991.doi:10.1016/0012-365X(91)90315-S MR1139449 (92j:11150) |

[3] | Harald Niederreiter. Point sets and sequences with small discrepancy. Monatshefte für Mathematik, 104(4):273–337, December 1987.doi:10.1007/BF01294651 MR918037 (89c:11120) |

[4] | 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 |

[5] | Gary L. Mullen, Arijit Mahalanabis, and Harald Niederreiter. Tables of ( t, m, s)-net and (t, s)-sequence parameters.In Harald Niederreiter and Peter Jau-Shyong Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106 of Lecture Notes in Statistics, pages 58–86. Springer-Verlag, 1995. |

### 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. “*k* = 2 Construction.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2015-09-03.
http://mint.sbg.ac.at/desc_NK2.html