Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation
Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any $h\geq2$ node erasures in a cooperative way, while retaining the minimum storage as an $[n,k]$ MDS code. Each code coordinate (node) is assumed to store an array of $\ell$ symbols, where $\ell$ is termed as sub-packetization. To address the disk IO (input/output) capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a lower bound on this quantity. In this paper, we focus on reducing the sub-packetization level of optimal-access cooperative MSR codes. We propose new constructions of optimal-access cooperative MSR codes through two methods. At first, we propose a direct explicit construction by designing its parity-check matrix. Such parity-check matrix is built by repeatedly employing two crucial parity-check matrices as building blocks. Secondly, we propose a generic transformation framework. Starting from an arbitrary $[n+d-k,d]$ MDS scalar code, one can derive a final cooperative MSR code by systematically applying two basic transformations. Both approaches yield $(n,k,\ell=\delta^m)$ optimal-access cooperative MSR codes with $\delta=d-k+h$ and $m=\binom{n}{h}-\lfloor\frac{n}{\delta}\rfloor(\binom{\delta}{h}-1)$. Compared with the state of the art (with $\ell=\delta^{\binom{n}{h}}$), the derived codes can reduce the sub-packetization $\ell$ by a fraction of $1/\delta^{\lfloor\frac{n}{\delta}\rfloor(\binom{\delta}{h}-1)}$, where $\delta=d-k+h$. Moreover, we also show that some previous code structures of optimal-access cooperative MSR codes and optimal-access MSR codes with $h=1$ are included as special cases of our transformation construction. At last, we note that all of the constructions are built over a finite field of linear size $\geq n+d-k$.