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 $$L_2$$ L 2 -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 $$L_2$$ L 2 -discrepancy for the derived two-level designs are built, and some new lower bounds of the wrap-around $$L_2$$ L 2 -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.

中文翻译：

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通过两次转换的四级和两级设计的新下界

近年来，代码理论被广泛用于构建优化设计。在本文中，考虑了两种变换，一种是修正的格雷图，一种是四级编码与四级设计的三个二进制码序列之间的映射。通过这两个变换，我们指出四水平设计对应的两水平设计的环绕$$L_2$$L 2 -差异值是由四水平设计决定的，两个新的解析表达式建立了派生的两级设计的环绕 $$L_2$$ L 2 -discrepancy，以及四级和两级环绕 $$L_2$$ L 2 -discrepancy 的一些新下界获得设计，可作为搜索统一设计和评价设计统一性的基准。此外，基于第二次变换，