text: 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.
  @cite_0: Maximum distance separable (MDS) codes are widely used in storage systems to protect against disk (node) failures. A node is said to have capacity l over some field F, if it can store that amount of symbols of the field. An (n, k, l) MDS code uses n nodes of capacity l to store k information nodes. The MDS property guarantees the resiliency to any n-k node failures. An optimal bandwidth (respectively, optimal access) MDS code communicates (respectively, accesses) the minimum amount of data during the repair process of a single failed node. It was shown that this amount equals a fraction of 1 (n - k) of data stored in each node. In previous optimal bandwidth constructions, l scaled polynomially with k in codes when the asymptotic rate is less than 1. Moreover, in constructions with a constant number of parities, i.e., when the rate approaches 1, l is scaled exponentially with k. In this paper, we focus on the case of linear codes with linear repair operations and constant number of parities n - k = r, and ask the following question: given the capacity of a node l what is the largest number of information disks k in an optimal bandwidth (respectively, access) (k + r, k, l) MDS code? We give an upper bound for the general case, and two tight bounds in the special cases of two important families of codes. The first is a family of codes with optimal update property, and the second is a family with optimal access property. Moreover, the bounds show that in some cases optimal-bandwidth codes have larger k than optimal-access codes, and therefore these two measures are not equivalent.
  @cite_8: The first focus of the present paper, is on lower bounds on the sub-packetization level @math of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on @math which is shown to be tight for the case @math by comparing with recent code constructions in the literature. We also extend our results to an @math MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level @math of an MDS code that can carry out repair of any node in a subset of @math nodes, @math where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on @math for the case of @math . This bound holds for any @math and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case @math Q @math $ . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code.
summary: In summary, while there are several constructions of high rate MSR codes, they all incur large sub-packetization, which is undesirable as briefly explained earlier. This has been partially explained by lower bounds on @math in a few previous works. For the special case of optimal-access MSR codes, a lower bound of @math was shown in @cite_0 , and this was improved (when all-node repair is desired) to @math recently @cite_8 . Together with the above-mentioned constructions, we thus have matching upper and lower bounds on @math for the optimal-access case. This help-by-transfer setting is primarily combinatorial in nature, which is exploited heavily in these lower bounds.