A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) block code. This tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T ≥ nt + nr − 1 where T is the number of time slots over which coding takes place and nt, nr are the number of transmit and receive antennas respectively. For T < nt + nr − 1, only upper and lower bounds on the D-MG tradeoff are available.In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T = nt = n, cyclic-division-algebra (CDA) based ST codes having the non-vanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n.For the rectangular, T > nt case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all T ≥ nt is the same as that previously known to hold for T ≥ nt + nr − 1.

Abstract-We address the problem of securing distributed storage systems against eavesdropping and adversarial attacks. An important aspect of these systems is node failures over time, necessitating, thus, a repair mechanism in order to maintain a desired high system reliability. In such dynamic settings, an important security problem is to safeguard the system from an intruder who may come at different time instances during the lifetime of the storage system to observe and possibly alter the data stored on some nodes. In this scenario, we give upper bounds on the maximum amount of information that can be stored safely on the system. For an important operating regime of the distributed storage system, which we call the bandwidthlimited regime, we show that our upper bounds are tight and provide explicit code constructions. Moreover, we provide a way to short list the malicious nodes and expurgate the system.

Abstract-In this paper, we quantify how much codes can reduce the data retrieval latency in storage systems. By combining a simple linear code with a novel request scheduling algorithm, which we call Blocking-one Scheduling (BoS), we show analytically that it is possible to reduce data retrieval delay by up to 17% over currently popular replication-based strategies. Although in this work we focus on a simplified setting where the storage system stores a single content, the methodology developed can be applied to more general settings with multiple contents. The results also offer insightful guidance to the design of storage systems in data centers and content distribution networks.

Given an n-length input signal x, it is well known that its Discrete Fourier Transform (DFT), X, can be computed from n samples in O(n log n) operations using a Fast Fourier Transform (FFT) algorithm. If the spectrum X is k-sparse (where k << n), can we do better? We show that asymptotically in k and n, when k is sub-linear in n (precisely, k = O(n δ ) where 0 < δ < 1), and the support of the non-zero DFT coefficients is uniformly random, our proposed FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm computes the DFT X, from O(k) samples in O(k log k) arithmetic operations, with high probability. Further, the constants in the big Oh notation for both sample and computational cost are small, e.g., when δ < 0.99, which essentially covers almost all practical cases of interest, the sample cost is less than 4k.Our approach is based on filterless subsampling of the input signal x using a set of carefully chosen uniform subsampling patterns guided by the Chinese Remainder Theorem (CRT). The idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by subsampling. Specifically, the subsampling operation on x is designed to create aliasing patterns on the spectrum X that "look like" parity-check constraints of a good erasure-correcting sparse-graph code. Next, we show that computing the sparse DFT X is equivalent to decoding of sparse-graph codes. These codes further allow for fast peeling-style decoding. The resulting DFT computation is low in both sample complexity and decoding complexity. We analytically connect our proposed CRT-based aliasing framework to random sparse-graph codes, and analyze the performance of our algorithm using density evolution techniques from coding theory. We also provide simulation results, that are in tight agreement with our theoretical findings.I. INTRODUCTION Spectral analysis using the Discrete Fourier Transform (DFT) has been of universal importance in engineering and scientific applications for a long time. The Fast Fourier Transform (FFT) is the fastest known way to compute the DFT of an arbitrary n-length signal, and has a computational complexity of O(n log n)1 . Many applications of interest involve signals, e.g. audio, image, video data, biomedical signals etc., which have a sparse Fourier spectrum. In such cases, a small subset of the spectral components typically contain most or all of the signal energy, with most spectral components being either zero or negligibly small. If the n-length DFT, X, is k-sparse, where k << n, can we do better in terms of both sample and computational complexity of computing the sparse DFT? We answer this question affirmatively. In particular, we show that asymptotically in k and n, when k is sub-linear in n (precisely, k = O(n δ ) where 0 < δ < 1), and the support of the non-zero DFT coefficients is uniformly random, our proposed FFAST (Fast Fourier Aliasing-based Sparse Transform) algorithm computes the DFT X, from judiciously chosen O(k) samples in O(k log k) arithmetic operations, with high pro...

Abstract-In this paper we study the data exchange problem where a set of users is interested in gaining access to a common file, but where each has only partial knowledge about it as side-information. Assuming that the file is broken into packets, the side-information considered is in the form of linear combinations of the file packets. Given that the collective information of all the users is sufficient to allow recovery of the entire file, the goal is for each user to gain access to the file while minimizing some communication cost. We assume that users can communicate over a noiseless broadcast channel, and that the communication cost is a sum of each user's cost function over the number of bits it transmits. For instance, the communication cost could simply be the total number of bits that needs to be transmitted. In the most general case studied in this paper, each user can have any arbitrary convex cost function. We provide deterministic, polynomial-time algorithms (in the number of users and packets) which find an optimal communication scheme that minimizes the communication cost. To further lower the complexity, we also propose a simple randomized algorithm inspired by our deterministic algorithm which is based on a random linear network coding scheme.