Combinatorial bounds and constructions in the theory of uniform point distributions in unit cubes, connections with orthogonal arrays and a poset generalization of a related problem in coding theory [microform]
- Bib ID:
- 3283562
- Format:
- Book and Microform
- Author:
- Lawrence, Kenneth Mark
- Description:
- 184 p.
- Summary:
-
We consider low-discrepancy point sets which are obtained from the theory of (t, m, s)-nets in base b, as introduced by Niederreiter in 1987. These nets are point sets in the s-dimensional unit cube $I\sp{s}$ = (0,1) $\sp{s}$ which show a high degree of uniformity in their distribution. They have applications to Quasi-Monte Carlo methods in numerical analysis and pseudorandom number generation. A central question is the determination of the parameter values for which (t, m, s)-nets exist. Our main purpose is to explore this central question, placing special emphasis on the link between (t, m, s)-nets and orthogonal arrays first observed by Niederreiter in 1992. Orthogonal arrays are basic combinatorial structures which have applications to experimental design and cryptology.
As a result of our investigations we obtain a geometrical equivalence between certain orthogonal arrays and the so-called "pseudo (t, m, s)-nets in base b", which are point sets in $I\sp{s}$ with somewhat weaker uniformity properties than those of (t, m, s)-nets. This equivalence generalizes Niederreiter's 1992 result, and leads to new accessary conditions for the existence of (t, m, s)-nets which, in general, are significantly more stringent than those previously known. We also define a new family of combinatorial objects called "generalized orthogonal arrays", and show a completely general equivalence between (t, m, s)-nets and the appropriate generalized orthogonal arrays. Using this combinatorial characterization we describe a new method for constructing (t, m, s)-nets. This method gives rise to some very general conditions on the parameters which are sufficient to ensure the existence of a (t, m, s)-net.
In this way we construct many nets which are apparently new. Finally, in 1987 Niederreiter considered a problem from combinatorial linear algebra which generalizes the following classical problem of coding theory: given a finite field $F\sb{q}$ and integers n $>$ k $\geq$ 1, find the largest minimum distance achievable by a linear code over $F\sb{q}$ of length n and dimension k. We place Niederreiter's problem in the more general setting of a poset (partially ordered set), P, by defining "poset-codes". The generalized problem may then be interpreted as the problem of finding the largest minimum P-distance attainable by a linear P-code over $F\sb{q}$ with fixed length and fixed dimension. We extend some of the bounds from Niederreiter's setting (where the poset is a disjoint union of chains) and also obtain bounds for posets which are the product of two chains.
- Notes:
-
- (UnM)AAI9527125
- Source: Dissertation Abstracts International, Volume: 56-07, Section: B, page: 3794.
- Supervisor: Richard A. Brualdi.
- Thesis (Ph.D.)--The University of Wisconsin - Madison, 1995.
- Reproduction:
- Microfiche. Ann Arbor, Mich.: University Microfilms International.
- Subject:
- Mathematics
- Other authors/contributors:
- The University of Wisconsin - Madison
- Copyright:
-
In Copyright
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Copyright status was determined using the following information:
- Material type:
- Literary Dramatic Musical
- Published status:
- Unpublished
- Creation date:
- 1995
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