An @math vector code @math is a collection of @math codewords where @math elements (from the field @math ) in each of the codewords are referred to as code blocks. Assuming that @math , the code blocks are treated as @math -length vectors over the base field @math . Equivalently, the code is said to have the sub-packetization level @math . This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code blocks. The repair bandwidth of a code measures the information flow from the remaining code blocks during the reconstruction of a single code block. This problem naturally arises in the context of distributed storage systems as the node repair problem [4]. Assuming that @math , the repair bandwidth of an MDS vector code is lower bounded by @math symbols (over the base field @math ) which is also referred to as the cut-set bound [4]. For all values of @math and @math , the MDS vector codes that attain the cut-set bound with the sub-packetization level @math are known in the literature [23, 35]. This paper presents a construction for MDS vector codes which simultaneously ensures both small repair bandwidth and small sub-packetization level. The obtained codes have the smallest possible sub-packetization level @math for an MDS vector code and the repair bandwidth which is at most twice the cut-set bound. The paper then generalizes this code construction so that the repair bandwidth of the obtained codes approach the cut-set bound at the cost of increased sub-packetization level. The constructions presented in this paper give MDS vector codes which are linear over the base field @math .
@cite_15: This paper considers a distributed storage system, where multiple storage nodes can be reconstructed simultaneously at a centralized location. This centralized multi-node repair (CMR) model is a generalization of regenerating codes that allow for bandwidth-efficient repair of a single failed node. This work focuses on the trade-off between the amount of data stored and repair bandwidth in this CMR model. In particular, repair bandwidth bounds are derived for the minimum storage multi-node repair (MSMR) and the minimum bandwidth multi-node repair (MBMR) operating points. The tightness of these bounds are analyzed via code constructions. The MSMR point is characterized through codes achieving this point under functional repair for general set of CMR parameters, as well as with codes enabling exact repair for certain CMR parameters. The MBMR point, on the other hand, is characterized with exact repair codes for all CMR parameters for systems that satisfy a certain entropy accumulation property. Finally, the model proposed here is utilized for the secret sharing problem, where the codes for the multi-node repair problem is used to construct communication efficient secret sharing schemes with the property of bandwidth efficient share repair.
Interestingly, the lower bound on the repair bandwidth of an MDS vector code given in continues to hold even when the contacted nodes contribute different number of symbols during the repair process @cite_15 , i.e., for every @math , we have