We investigate $p$-ary $t$-designs which are simultaneously designs for all $t$, which we call universal $p$-ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non-null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs, up to similarity.

中文翻译：

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通用 p-ary 设计

我们调查 $磷$-ary $吨$-同时为所有人设计的设计 $吨$，我们称之为普遍 $磷$-ary 设计。由于戈登詹姆斯通过对称群的表示理论，零通用设计被很好地理解。我们研究非零设计并确定存在此类设计的系数的充分必要条件。这使我们能够对所有通用设计进行分类，直至相似。