MDS Code Constructions with Small Sub-packetization and Near-optimal Repair Bandwidth

A code C ⊆ F is a collection of M codewords where n elements (from the finite field F) in each of the codewords are referred to as code blocks. Assuming that F is a degree ` extension of a smaller field B, the code blocks are treated as `-length vectors over the base field B. Equivalently, the code is said to have the sub-packetization level `. This paper addresses the problem of constructing MDS codes that enable exact reconstruction (repair) of each code block by downloading 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. The problem of enabling exact reconstruction of a code block with small repair bandwidth naturally arises in the context of distributed storage systems as the node repair problem [7]. The constructions of exact-repairable MDS codes with optimal repair-bandwidth require working with large sub-packetization levels, which restricts their employment in practice. This paper presents constructions for MDS codes that simultaneously provide both small repair bandwidth and small sub-packetization level. In particular, this paper presents two general approaches to construct exact-repairable MDS codes that aim at significantly reducing the required sub-packetization level at the cost of slightly sub-optimal repair bandwidth. The first approach provides MDS codes that have repair bandwidth at most twice the optimal repair-bandwidth. Additionally, these codes also have the smallest possible sub-packetization level ` = O(r), where r 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 sub-packetization level. The second approach provides ways to transform an MDS code with optimal repair-bandwidth and large sub-packetization level into a longer MDS code with small sub-packetization level and near-optimal repair bandwidth. For a given number of parity blocks, the codes constructed using this approach have their sub-packetization level scaling logarithmically This research was supported in part by NSF grants CCF-0963975, CCF-1422045, and CCF-1563742. This paper was presented in parts at the 2016 Information Theory and Applications Workshop [35], 2017 ACM-SIAM Symposium on Discrete Algorithms [13] and 2017 IEEE International Symposium on Information Theory [29]. A. S. Rawat is with the Research Laboratory of Electronics, MIT, Cambridge, MA 02139, USA (e-mail: [email protected]). Part of this work was done when the author was with the Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. I. Tamo is with the Department of Electrical Engineering Systems, Tel Aviv University, Ramat Aviv 69978, Israel (e-mail: [email protected]). V. Guruswami is with the Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: [email protected]). K. Efremenko is with the Department of Computer Science, Tel Aviv University, Ramat Aviv 69978, Israel (e-mail: [email protected]). September 26, 2017 DRAFT ar X iv :1 70 9. 08 21 6v 1 [ cs .I T ] 2 4 Se p 20 17