Explicit Constructions of High-Rate MSR Codes with Optimal Access Property over Small Finite Fields

Abstract

Up to now, many $(k+r,k,N)$ minimum-storage regenerating (MSR) codes with $k$ information nodes, $r$ parity nodes, and node capacity $N$ have been proposed. However, most of them are constructed over a relatively large finite field. In this paper, we propose three high-rate MSR codes over small finite fields. First, the new MSR code $\mathcal {C}_{1}$ with the optimal access property for all nodes is constructed over small finite field $\mathbb {F}_{q}$ , for example $q=3$ for even $r$ or $q\ge r+1$ for odd $r$ , which is much smaller than that of the known one given by Ye and Barg. Further, considering to reduce the node capacity, another new MSR code $\mathcal {C}_{2}$ over $\mathbb {F}_{q}$ with $q\ge r+2$ is generated based on $\mathcal {C}_{1}$ , which can effectively reduce the node capacity of $\mathcal {C}_{1}$ by a factor of $r^{r-1}$ . However, only the first $k$ nodes of $\mathcal {C}_{2}$ have the optimal access property. Therefore, the new MSR code $\mathcal {C}_{3}$ over $\mathbb {F}_{q}$ with $q\ge r+2$ which has the optimal access property for all nodes is proposed by modifying $\mathcal {C}_{2}$ . Notably, in contrast to $\mathcal {C}_{1}$ , the node capacity of $\mathcal {C}_{3}$ is decreased by a factor of $r^{r-2}$ .