Universal $p$-ary designs

Abstract

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.