Explicit Construction of Minimum Storage Rack-Aware Regenerating Codes for All Parameters

We consider the rack-aware storage system where $n=\bar{n}u$ nodes are organized in $\bar{n}$ racks each containing $u$ nodes, and any $k=\bar{k}u+u_0~(0\leq u_0<u)$ nodes can retrieve the original data file. More importantly, the cross-rack communication cost is much more expensive than the intra-rack communication cost, so that the latter is usually neglected in the system bandwidth. The MSRR (minimum storage rack-aware regenerating) code is an important variation of regenerating codes that achieves the optimal repair bandwidth for single node failures in the rack-aware model. However, explicit construction of MSRR codes for all parameters were not developed until Chen\&Barg's work. In this paper we present another explicit construction of MSRR codes for all parameters that improve Chen\&Barg's construction in two aspects: (1) The sub-packetization is reduced from $(\bar{d}-\bar{k}+1)^{\bar{n}}$ to $(\bar{d}-\bar{k}+1)^{\lceil\frac{\bar{n}}{u-u_{0}}\rceil}$ where $\bar{d}$ is the number of helper racks that participate in the repair process; (2) The field size is reduced to $|F|>n$ which is almost half of the field used in Chen\&Barg's construction. Besides, our code keeps the same access level as Chen\&Barg's low-access construction.