Universal $p$-ary designs

TL;DR

This paper studies universal $p$-ary $t$-designs that work for all $t$, providing a classification of such designs by analyzing their coefficients and extending understanding beyond null designs.

Contribution

It characterizes necessary and sufficient conditions for the existence of non-null universal $p$-ary $t$-designs and classifies all such designs up to similarity.

Findings

Necessary and sufficient conditions for non-null universal designs.

Complete classification of universal $p$-ary $t$-designs.

Extension of known results from null designs to all 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.