On MSR Subspace Families of Lines

A minimum storage regenerating (MSR) subspace family of ${\mathbb{F}}_q^{2m}$ is a set $\mathcal{S}$ of $m$-spaces in ${\mathbb{F}}_q^{2m}$ such that for any $m$-space $S$ in $\mathcal{S}$ there exists an element in $\mathrm{P}\mathrm{G}\mathrm{L}(2m,q)$ which maps $S$ to a complement and fixes $\mathcal{S}\setminus \{S\}$ pointwise. We show that an MSR subspace family of $2$-spaces in ${\mathbb{F}}_q^4$ has at most size $6$ with equality if and only if it is a particular subset of a Segre variety. This implies that an $(n,n-2,4)$-MSR code has $n\le 9$.