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Updated tables of parameters of \((t,m,s)\)-nets. (English) Zbl 0929.05011

This paper is devoted to updating the tables for \((t,m,s)\)-nets in base \(b\) for \(b \in \{2,3,5\}\). Besides, there are excellent discussions on constructions of \((t,m,s)\)-nets and \((t,s)\)-sequences and on nonexistence bounds. Ten constructions of nets are reported to be developed recently. This increases the number of known approaches to twenty-six. There are good references to all constructions under discussion. The authors also discuss the upper bounds on \(s\) (which correspond to lower bounds on \(t\) when \(m\), \(s\) and \(b\) are prescribed). The main tool for obtaining such bounds is based on the relations between the \((t,m,s)\)-nets and corresponding generalized orthogonal arrays. In particular, this allows using the powerful linear programming approach which often (but not always) gives the best known bounds. Tables are provided for the possible values of \(t\) and \(s\) for bases \(b=2\), 3 and 5. (Actually, in the paper the reader can find tables for \(b=2\) only; others are posted on JCD web page with the intention to be further updated.) The \(t\) tables cover (for each base) the range \(1 \leq m,s \leq 50\) and the \(s\) tables cover \(0 \leq t \leq 50\) and \(1 \leq m \leq 50\). Summarizing, this nice paper contains not only the updated tables but discussions on \((t,m,s)\)-nets and \((t,s)\)-sequences with many references.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
05B30 Other designs, configurations

Software:

AMPL; Maple; CPLEX
PDFBibTeX XMLCite
Full Text: DOI

References:

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