Additivity of symmetric and subspace designs

TL;DR

This paper investigates the conditions under which symmetric and subspace 2-designs can be embedded in abelian groups, establishing new results on their additivity and strong additivity properties.

Contribution

It provides explicit results on the additivity of symmetric and subspace 2-designs, including the proof of strong additivity of PG$_d(n,q)$ for all q.

Findings

Strong additivity of PG$_d(n,q)$ for all q

Additivity conditions for symmetric designs

Additivity of subspace 2-designs

Abstract

A $2$-$(v,k,\lambda)$ design is additive (or strongly additive) if it is possible to embed it in a suitable abelian group $G$ in such a way that its block set is contained in (or coincides with) the set of all the zero-sum $k$-subsets of $G$. Explicit results on the additivity or strong additivity of symmetric designs and subspace 2-designs are presented. In particular, the strong additivity of PG$_d(n,q)$, which was known to be additive only for $q=2$ or $d=n-1$, is always established.