An @math -vector MDS code is a @math -linear subspace of @math (for some field @math ) of dimension @math , such that any @math (vector) symbols of the codeword suffice to determine the remaining @math (vector) symbols. The length @math of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading @math field elements (which is known to be the least possible) from each of the other symbols. MSR codes are attractive for use in distributed storage systems, and by now a variety of ingenious constructions of MSR codes are available. However, they all suffer from exponentially large sub-packetization @math . Our main result is an almost tight lower bound showing that for an MSR code, one must have @math . Previously, a lower bound of @math , and a tight lower bound for a restricted class of "optimal access" MSR codes, were known. Our work settles a central open question concerning MSR codes that has received much attention. Further our proof is really short, hinging on one key definition that is somewhat inspired by Galois theory.
Regenerating and MSR codes have close connections to communication-efficient secret sharing schemes, which were studied and developed in @cite_12 . In this context, the sub-packetization corresponds to the size of the shares that the parties must hold.
@cite_12
2007636263
In this paper, we consider the security of exact-repair regenerating codes operating at the minimum-storage-regenerating (MSR) point. The security requirement (introduced in Shah et. al.) is that no information about the stored data file must be leaked in the presence of an eavesdropper who has access to the contents of l 1 nodes as well as all the repair traffic entering a second disjoint set of l 2 nodes. We derive an upper bound on the size of a data file that can be securely stored that holds whenever l 2 ≤ d - k + 1. This upper bound proves the optimality of the product-matrix-based construction of secure MSR regenerating codes by Shah et. al.