New lower bounds of four-level and two-level designs via two transformations

Hongyi Li and Hong Qin ()
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Hongyi Li: Jishou University
Hong Qin: Central China Normal University

Statistical Papers, 2020, vol. 61, issue 3, No 15, 1243 pages

Abstract: Abstract Code theory is widely used to construct optimal designs in recent years. In this paper, two transformations, a modified Gray map and a mapping between quaternary codes and the sequence of three binary codes for four-level designs, are considered. Via the two transformations, we point out that the wrap-around $$L_2$$L2-discrepancy values of the two-level designs corresponding to a four-level design are decided by the four-level design, two new analytical expressions of the wrap-around $$L_2$$L2-discrepancy for the derived two-level designs are built, and some new lower bounds of the wrap-around $$L_2$$L2-discrepancy for four-level and two-level designs are obtained, which can be used as a benchmark for search the uniform designs and evaluate the uniformity of designs. Furthermore, based on the second transformation, we provide a very strong link between the aberration of a four-level design and the uniformity of the derived two-level design.

Keywords: Modified Gray map; Lower bound; Quaternary code; Uniform design; Wrap-around $$L_2$$ L 2 -discrepancy; 62K15; 62K05; 62K99 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s00362-018-0987-z

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