Zero-sum flows for Steiner systems

Given a $t$-$(v, k, lambda)$ design, $mathcal{D}=(X,mathcal{B})$, a zero-sum $n$-flow of $mathcal{D}$ is a map $f : mathcal{B}longrightarrow {pm1,ldots, pm(n-1)}$ such that for any point $xin X$, the sum of $f$ over all blocks incident with $x$ is zero. For a positive integer $k$, we find a zero-sum $k$-flow for an STS$(u w)$ and for an STS$(2v+7)$ for $vequiv 1~(mathrm{mod}~4)$, if there are STS$(u)$, STS$(w)$ and STS$(v)$ such that the STS$(u)$ and STS$(v)$ both have a zero-sum $k$-flow. In 2015, it was conjectured that for $v>7$ every STS$(v)$ admits a zero-sum $3$-flow. Here, it is shown that many cyclic STS$(v)$ have a zero-sum $3$-flow. Also, we investigate the existence of zero-sum flows for some Steiner quadruple systems.