Augmented designs are typically used in early-stage breeding programs to compare single replicates of test entries by combining them with replicated check varieties. One or two dimensional incomplete blocking can be incorporated in the design to accommodate possible site variation. An augmented design in a square array can be derived from a smaller row-column design (the contraction). In a recent paper Bailey and Haines (2025) investigated the link between an augmented design in a square array and its contraction. Here we formally establish this connection by expressing the average efficiency factor of the augmented design in terms of that of its contraction. A consequence of this is that an optimal contraction can be used to construct an optimal augmented design. The table of cyclic contractions presented by Bailey and Haines (2025) is updated in terms of optimality. Specifically, in cases where a cyclic contraction is not optimal, an augmented design with optimal or near-optimal efficiency can be obtained via computer search.

翻译：增强设计通常用于早期育种计划，通过将单次重复的测试条目与重复的对照品种相结合进行比较。设计中可纳入一维或二维不完全区组以应对可能的场地变异。方形阵列中的增强设计可从较小的行-列设计（收缩设计）推导得出。在近期的一篇论文中，Bailey和Haines（2025）研究了方形阵列中增强设计与其收缩设计之间的关联。本文通过将增强设计的平均效率因子以其收缩设计的效率因子表达，正式建立了这一联系。由此推论，最优收缩设计可用于构建最优增强设计。Bailey和Haines（2025）提出的循环收缩设计表已根据最优性准则更新。具体而言，当循环收缩设计非最优时，可通过计算机搜索获得具有最优或接近最优效率的增强设计。