Scalar MSCR Codes via the Product Matrix Construction

An $(n,k,d)$ cooperative regenerating code provides the optimal-bandwidth repair for any $t~(t!>!1)$ node failures in a cooperative way. In particular, an MSCR (minimum storage cooperative regenerating) code retains the same storage overhead as an $(n,k)$ MDS code. Suppose each node stores $\alpha$ symbols which indicates the sub-packetization level of the code. A scalar MSCR code attains the minimum sub-packetization, i.e., $\alpha=d-k+t$. By now, all existing constructions of scalar MSCR codes restrict to very special parameters, eg. $d=k$ or $k=2$, etc. In a recent work, Ye and Barg construct MSCR codes for all $n,k,d,t$, however, their construction needs $\alpha\approx{\rm exp}(n^t)$ which is almost infeasible in practice. In this paper, we give an explicit construction of scalar MSCR codes for all $d\geq \max{2k-1-t,k}$, which covers all possible parameters except the case of $k\leq d\leq 2k-2-t$ when $k<2k-1-t$. Moreover, as a complementary result, for $k<d<2k-2-t$ we prove the nonexistence of linear scalar MSCR codes that have invariant repair spaces. Our construction and most of the previous scalar MSCR codes all have invariant repair spaces and this property is appealing in practice because of convenient repair. As a result, this work presents an almost full description of linear scalar MSCR codes.