Explicit Construction of ($k+r, k$) MDS Code with Small Sub-packetization level and Optimal Access Property for All Nodes

In practical distributed storage systems, the sub-packetization level of MDS codes is required as small as possible. In the literature, by sacrificing the optimality of the repair bandwidth, some constructions of MDS codes with small sub-packetization level were presented in the work of Rawat et al. (IEEE Trans. Inform. Theory, 64(10), 6506–6525, Oct. 2018) and Li et al. (IEEE Trans. Inform. Theory, 67(4), 2162–2180, Apr. 2021). However, the sub-packetization level of those constructions given in the two works are the power of the number of the parity nodes, which is still relatively large for the practical systems. In this paper, we construct an MDS code over a small finite field, whose sub-packetization level can be the power of a positive integer $s$ , where $s$ is less than the number of the parity nodes.