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.
Turning to known lower bounds on @math , a weak bound of @math was shown via a combinatorial argument in @cite_0 . Using an elegant linear independence and partitioning argument, the following bound is proven in @cite_24 : (This was slightly improved in @cite_10 , but the improvement is tiny for the case when @math which is our focus.) The bound implies a lower bound on sub-packetization of @math . Even for the case @math , it was not known if one can achieve sub-packetization smaller than @math . Our Theorem now rules out this possibility. We conjecture that our bound can be improved to @math which will show that the construction in @cite_15 is exactly tight.
@cite_0 @cite_15 @cite_10 @cite_24
2765728635 2963333438 30524954 2951592515
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 . We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of @math nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level @math . It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code. 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. In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by @math or @math in the polynomial ring @math , @math are coprime, which is called over @math . The immunity of @math is lower bounded by @math , which is achievable when @math is a multiple of @math ; the immunity of @math is exactly @math for every @math and @math . Our result improves the previous bound @math by Green. We observe how immunity over @math is related to @math circuit lower bound. For example, if the immunity of @math over @math is lower bounded by @math , and @math , then @math requires @math circuit of exponential size to compute. This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function @math with respect to a distribution @math is @math , then any two party randomized protocol computing @math must reveal at least @math bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on @math must reveal @math bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is @math , which provides an alternative proof to the recent proof of Viola Viola11 of the @math lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight @math lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of kerenidis2012lower , this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.