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.
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  @cite_1: Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An [n, k, d] regenerating code permits the data to be recovered by connecting to any k of the n nodes in the network, while requiring that a failed node be repaired by connecting to any d nodes. The amount of data downloaded for repair is typically much smaller than the size of the source data. Previous constructions of exact-regenerating codes have been confined to the case n=d+1 . In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of [n, k, d] and (b) Minimum Storage Regenerating (MSR) codes for all [n, k, d ≥ 2k-2], using a new product-matrix framework. The product-matrix framework is also shown to significantly simplify system operation. To the best of our knowledge, these are the first constructions of exact-regenerating codes that allow the number n of nodes in the network, to be chosen independent of the other parameters. The paper also contains a simpler description, in the product-matrix framework, of a previously constructed MSR code with [n=d+1, k, d ≥ 2k-1].
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  @cite_5: The high repair bandwidth cost of (n,k) maximum distance separable (MDS) erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional MDS codes. In this paper, we address (n,k,d) exact repair MDS codes, which allow for any single failed node to be repaired exactly with access to any arbitrary set of d survivor nodes. We show the existence of exact repair MDS codes that achieve minimum repair bandwidth (matching the cut-set lower bound) for arbitrary admissible (n,k,d), i.e., k ≤ d ≤ n-1. Moreover, we extend our results to show the optimality of our codes for multiple-node failure scenarios in which an arbitrary set of r ≤ n-k failed nodes needs to repaired. Our approach is based on asymptotic interference alignment proposed by Cadambe and Jafar. As a byproduct, we also characterize the capacity of a class of multisource nonmulticast networks.
  @cite_15: MDS codes are erasure-correcting codes that can correct the maximum number of erasures given the number of redundancy or parity symbols. If an MDS code has r parities and no more than r erasures occur, then by transmitting all the remaining data in the code one can recover the original information. However, it was shown that in order to recover a single symbol erasure, only a fraction of 1 r of the information needs to be transmitted. This fraction is called the repair bandwidth (fraction). Explicit code constructions were given in previous works. If we view each symbol in the code as a vector or a column, then the code forms a 2D array and such codes are especially widely used in storage systems. In this paper, we ask the following question: given the length of the column l, can we construct high-rate MDS array codes with optimal repair bandwidth of 1 r, whose code length is as long as possible? In this paper, we give code constructions such that the code length is (r + l)log r l.
  @cite_16: We present a high-rate (n, k, d = n − 1)-MSR code with a sub-packetization level that is polynomial in the dimension k of the code. While polynomial sub-packetization level was achieved earlier for vector MDS codes that repair systematic nodes optimally, no such MSR code construction is known. In the low-rate regime (i. e., rates less than one-half), MSR code constructions with a linear sub-packetization level are available. But in the high-rate regime (i. e., rates greater than one-half), the known MSR code constructions required a sub-packetization level that is exponential in k. In the present paper, we construct an MSR code for d = n − 1 with a fixed rate equation, achieveing a sub-packetization level α = O(kt). The code allows help-by-transfer repair, i. e., no computations are needed at the helper nodes during repair of a failed node.
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  @cite_11: In distributed storage systems that employ erasure coding, the issue of minimizing the total communication required to exactly rebuild a storage node after a failure arises. This repair bandwidth depends on the structure of the storage code and the repair strategies used to restore the lost data. Designing high-rate maximum-distance separable (MDS) codes that achieve the optimum repair communication has been a well-known open problem. Our work resolves, in part, this open problem. In this study, we use Hadamard matrices to construct the first explicit two-parity MDS storage code with optimal repair properties for all single node failures, including the parities. Our construction relies on a novel method of achieving perfect interference alignment over finite fields with a finite number of symbol extensions. We generalize this construction to design @math -parity MDS codes that achieve the optimum repair communication for single systematic node failures.
summary: We begin code constructions existence results. present an explicit construction of MSR codes with small sub-packetization @math when the code rate @math is at most @math @cite_1 . @cite_5 show the existence of high rate MSR codes when the sub-packetization approaches infinity. Motivated by this result, the problem of designing high-rate MSR codes with finite sub-packetization level is explored in @cite_11 @cite_27 @cite_26 @cite_23 @cite_14 @cite_28 @cite_2 @cite_13 @cite_4 and references therein. In particular, @cite_16 show the existence of MSR codes with the sub-packetization level @math . Such a result with similar sub-packetization levels for repair of only @math systematic nodes was obtained earlier in @cite_15 @cite_14 . In order to ensure the MDS property, these results relied on huge fields and randomized construction of the parity check matrices.