MDS Code Constructions With Small Sub-Packetization and Near-Optimal Repair Bandwidth

This paper addresses the problem of constructing maximum distance separable (MDS) codes that enable exact reconstruction (repair) of each code block by downloading a small amount of information from the remaining code blocks. The total amount of information flow from the remaining code blocks during this reconstruction process is referred to as repair bandwidth of the underlying code. Existing constructions of exact-repairable MDS codes with optimal repair bandwidth require working with large subpacketization levels, which restrict their applicability in practice. This paper presents two general approaches to construct exact-repairable MDS codes that aim at significantly reducing the required subpacketization level at the cost of slightly suboptimal repair bandwidth. The first approach provides MDS codes that have repair bandwidth at most twice the optimal repair bandwidth. In addition, these codes also have the smallest possible subpacketization level <inline-formula> <tex-math notation="LaTeX">$O(r)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> denotes the number of parity blocks. This approach is then generalized to design codes that have their repair bandwidth approaching the optimal repair bandwidth at the cost of graceful increment in the required subpacketization level. The second approach transforms an MDS code with optimal repair bandwidth and large subpacketization level into a longer MDS code with small subpacketization level and near-optimal repair bandwidth. For a given <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>, the codes constructed using this approach have their subpacketization level scaling logarithmically with the code length. In addition, the obtained codes require field size only linear in the code length and ensure load balancing among the intact code blocks in terms of the information downloaded from these blocks during the exact reconstruction of a code block.