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)}$suchthatforanypoint$x\in X$,thesumof$f$overallblocksincidentwith$x$iszero\mathrm{.}Forapositiveinteger$k$, we find a zero-sum $k$-flow for an STS$(uw)$ and for an STS$(2v+7)$ for $v\equiv 1~($\mathrm{m}\mathrm{o}\mathrm{d}$~4)$,ifthereareSTS$(u)$,STS$(w)$andSTS$(v)$ 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.