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 $\mathrm{PGL}(2m, q)$ which maps $S$ to itself 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 $n \leq 9$.

翻译：一个最小存储再生（MSR）子空间族是$\mathbb{F}_q^{2m}$中的一组$m$维子空间$\mathcal{S}$，使得对于$\mathcal{S}$中的任意$m$维子空间$S$，存在$\mathrm{PGL}(2m, q)$中的一个元素将$S\)映射到自身，并逐点固定$\mathcal{S} \setminus \{ S \}$。我们证明，$\mathbb{F}_q^4$中$2$维子空间的MSR子空间族至多包含$6$个元素，且达到该上界当且仅当该族是Segre簇的一个特定子集。这意味着$(n, n-2, 4)$-MSR码满足$n \leq 9$。