On Distributed Storage Allocations for Memory-Limited Systems

In this paper we consider distributed allocation problems with memory constraint limits. Firstly, we propose a tractable relaxation to the problem of optimal symmetric allocations from [1]. The approximated problem is based on the Q-error function, and its solution approaches the solution of the initial problem, as the number of storage nodes in the network grows. Secondly, exploiting this relaxation, we are able to formulate and to solve the problem for storage allocations for memory-limited DSS storing and arbitrary memory profiles. Finally, we discuss the extension to the case of multiple data objects, stored in the DSS.Comment: Submitted to IEEE GLOBECOM'1

On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes

This paper addresses the issue of design of low-rate sparse-graph codes with linear minimum distance in the blocklength. First, we define a necessary condition which needs to be satisfied when the linear minimum distance is to be ensured. The condition is formulated in terms of degree-1 and degree-2 variable nodes and of low-weight codewords of the underlying code, and it generalizies results known for turbo codes [8] and LDPC codes. Then, we present a new ensemble of low-rate codes, which itself is a subclass of TLDPC codes [4], [5], and which is designed under this necessary condition. The asymptotic analysis of the ensemble shows that its iterative threshold is situated close to the Shannon limit. In addition to the linear minimum distance property, it has a simple structure and enjoys a low decoding complexity and a fast convergence.Comment: submitted to IEEE Trans. on Communication

Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) code ensembles defined over the general linear group on the binary erasure channel. In particular, we prove threshold saturation of belief propagation decoding to the so called potential threshold, using the proof technique based on potential functions introduced by Yedla \textit{et al.}, assuming that the potential function exists. We rewrite the density evolution of nonbinary SC-LDPC codes in an equivalent vector recursion form which is suited for the use of the potential function. We then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials, and give a method to construct it. We define a potential function in a slightly more general form than one by Yedla \textit{et al.}, in order to make the technique based on potential functions applicable to the case of nonbinary LDPC codes. We show that the potential function exists if a solution to a carefully designed system of linear equations exists. Furthermore, we show numerically the existence of a solution to the system of linear equations for a large number of nonbinary LDPC code ensembles, which allows us to define their potential function and thus prove threshold saturation.Comment: To appear in IT Transaction

New constructions of CSS codes obtained by moving to higher alphabets

We generalize a construction of non-binary quantum LDPC codes over \F_{2^m} due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed $m$.Comment: 9 pages, 9 figures, full version of a paper submitted to the IEEE Symposium on Information Theor

Proving Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

We analyze nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group for transmission over the binary erasure channel. We prove threshold saturation of the belief propagation decoding to the potential threshold, by generalizing the proof technique based on potential functions recently introduced by Yedla et al.. The existence of the potential function is also discussed for a vector sparse system in the general case, and some existence conditions are developed. We finally give density evolution and simulation results for several nonbinary SC-LDPC code ensembles.Comment: in Proc. 2014 XXXIth URSI General Assembly and Scientific Symposium, URSI GASS, Beijing, China, August 16-23, 2014. Invited pape

Repair Scheduling in Wireless Distributed Storage with D2D Communication

We consider distributed storage (DS) for a wireless network where mobile devices arrive and depart according to a Poisson random process. Content is stored in a number of mobile devices, using an erasure correcting code. When requesting a piece of content, a user retrieves the content from the mobile devices using device-to-device communication or, if not possible, from the base station (BS), at the expense of a higher communication cost. We consider the repair problem when a device that stores data leaves the network. In particular, we introduce a repair scheduling where repair is performed (from storage devices or the BS) periodically. We derive analytical expressions for the overall communication cost of repair and download as a function of the repair interval. We illustrate the analysis by giving results for maximum distance separable codes and regenerating codes. Our results indicate that DS can reduce the overall communication cost with respect to the case where content is only downloaded from the BS, provided that repairs are performed frequently enough. The required repair frequency depends on the code used for storage and the network parameters. In particular, minimum bandwidth regenerating codes require very frequent repairs, while maximum distance separable codes give better performance if repair is performed less frequently. We also show that instantaneous repair is not always optimal.Comment: To be presented at IEEE Information Theory Workshop (ITW) 2015, Jeju Island, Korea, October 201

A Family of Erasure Correcting Codes with Low Repair Bandwidth and Low Repair Complexity

We present the construction of a new family of erasure correcting codes for distributed storage that yield low repair bandwidth and low repair complexity. The construction is based on two classes of parity symbols. The primary goal of the first class of symbols is to provide good erasure correcting capability, while the second class facilitates node repair, reducing the repair bandwidth and the repair complexity. We compare the proposed codes with other codes proposed in the literature.Comment: Accepted, will appear in the proceedings of Globecom 2015 (Selected Areas in Communications: Data Storage

Distributed Storage in Mobile Wireless Networks with Device-to-Device Communication

We consider the use of distributed storage (DS) to reduce the communication cost of content delivery in wireless networks. Content is stored (cached) in a number of mobile devices using an erasure correcting code. Users retrieve content from other devices using device-to-device communication or from the base station (BS), at the expense of higher communication cost. We address the repair problem when a device storing data leaves the cell. We introduce a repair scheduling where repair is performed periodically and derive analytical expressions for the overall communication cost of content download and data repair as a function of the repair interval. The derived expressions are then used to evaluate the communication cost entailed by DS using several erasure correcting codes. Our results show that DS can reduce the communication cost with respect to the case where content is downloaded only from the BS, provided that repairs are performed frequently enough. If devices storing content arrive to the cell, the communication cost using DS is further reduced and, for large enough arrival rate, it is always beneficial. Interestingly, we show that MDS codes, which do not perform well for classical DS, can yield a low overall communication cost in wireless DS.Comment: After final editing for publication in TCO