Locally Encodable and Decodable Codes for Distributed Storage Systems

We consider the locality of encoding and decoding operations in distributed storage systems (DSS), and propose a new class of codes, called locally encodable and decodable codes (LEDC), that provides a higher degree of operational locality compared to currently known codes. For a given locality structure, we derive an upper bound on the global distance and demonstrate the existence of an optimal LEDC for sufficiently large field size. In addition, we also construct two families of optimal LEDC for fields with size linear in code length.Comment: 7 page

Matrix Multiplication Verification Using Coding Theory

We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three $n \times n$ matrices $A$, $B$, and $C$ as input, to decide whether $AB = C$. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in $\widetilde{O}(n^2)$ time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in $o(n^{\omega})$ time). To that end, we give two algorithms for MMV in the case where $AB - C$ is sparse. Specifically, when $AB - C$ has at most $O(n^{\delta})$ non-zero entries for a constant $0 \leq \delta < 2$, we give (1) a deterministic $O(n^{\omega - \varepsilon})$-time algorithm for constant $\varepsilon = \varepsilon(\delta) > 0$, and (2) a randomized $\widetilde{O}(n^2)$-time algorithm using $\delta/2 \cdot \log_2 n + O(1)$ random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when $\delta \geq 1.056$, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses $\log_2 n + O(1)$ random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require $\Omega(n^{\omega})$ time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic $\widetilde{O}(n^2)$-time reductions)

Network Coding for Wireless and Wired Networks: Design, Performance and Achievable Rates

Many point-to-point communication problems are relatively well understood in the literature. For example, in addition to knowing what the capacity of a point-to-point channel is, we also know how to construct codes that will come arbitrarily close to the capacity of these channels. However, we know very little about networks. For example, we do not know the capacity of the two-way relay channel which consists of only three transmitters. The situation is not so different in the wired networks except special cases like multicasting. To understand networks better, in this thesis we study network coding which is considered to be a promising technique since the time it was shown to achieve the single-source multicast capacity. First we design and analyze deterministic and random network coding schemes for a cooperative communication setup with multiple sources and destinations. We show that our schemes outperform conventional cooperation in terms of the diversity-multiplexing tradeoff (DMT). Specifically, it can offer the maximum diversity order at the expense of a slightly reduced multiplexing rate. We derive the necessary and sufficient conditions to achieve the maximum diversity order. We show that when the parity-check matrix for a systematic maximum distance separable (MDS) code is used as the network coding matrix, the maximum diversity is achieved. We present two ways to generate full-diversity network coding matrices: namely using the Cauchy matrices and the Vandermonde matrices. We also analyze a selection relaying scheme and prove that a multiplicative diversity order is possible with enough number of relay selection rounds. In addition to the above scheme for wireless networks, we also study wired networks, and apply network coding together with interference alignment. We consider networks with $K$ source nodes and $J$ destination nodes with arbitrary message demands. We first consider a simple network consisting of three source nodes and four destination nodes and show that each user can achieve a rate of one half. Then we extend the result for the general case which states that when the min-cut between each source-destination pair is one, it is possible to achieve a sum rate that is arbitrarily close to the min-cut between the source nodes whose messages are demanded and the destination node where the sum rate is the summation of all the demanded source message rates plus the biggest interferer\u27s rate

Flexible Cross-Subspace Alignment Codes for Variable Coded Distributed Batch Matrix Multiplication

Modern distributed systems suffer from a phenomenon known as stragglers where computation nodes either break-down or are sufficiently slow, resulting in a large tail latency. Inspired by error correcting codes, researchers within the field of coded computation combat stragglers by cleverly encoding the data within the computations. One major endeavor is in the study of coded matrix-matrix multiplication where the task is to multiply two large matrices in a distributed manner. Most coded matrix computation work focuses on highly structured tasks which allows for easier code construction and analysis but limits the applicability for more general problems. For the first time, we consider the novel problem of multiplying many different matrices whose products may share matrices with no guaranteed regularity. Specifically, we consider the Variable Coded Distributed Batch Matrix Multiplication (VCDBMM) problem where the system is given two sets of matrices $\mathcal{A}=\{A_1,A_2,\dots,A_{|\mathcal{A}|}\}$ and $\mathcal{B}=\{B_1,B_2,\dots,B_{|\mathcal{B}|}\}$ and a set of computation goals $\mathcal{S} = \{(i_1,j_1),(i_2,j_2), \dots (i_{|\mathcal{S}|},j_{|\mathcal{S}|})\}$ and the objective is to calculate the matrix multiplication $A_iB_j$ for every $(i,j) \in \mathcal{S}$ in the presence of stragglers. Therefore, a good coding solution minimizes the recovery threshold (i.e., the number of workers that we need to wait for in order to compute the final output). Inspired by Cross-Subspace Alignment Codes, we construct Flexible Cross-Subspace Alignment Codes (FCSA) to solve the general VCDBMM problem. We provide two variants of FCSA codes that allow for a trade-off between the encoding/decoding complexity and the recovery threshold. We demonstrate that both variants are within a factor of two optimal under certain system constraints. We also generalize FCSA codes into Grouped FCSA codes where we group computations together to provide further flexibility between the computational complexity at the workers and the recovery threshold. We provide simulations on random instances of the VCDBMM problem and demonstrate the average improvement offered by our codes

A construction of MDS array codes

In this paper a new construction of MDS array codes is introduced. In order to obtain a code with this property, the parity-check matrix is constructed just using a superregular matrix of blocks composed of powers of the companion matrix of a primitive polynomial. Also a decoding algorithm for these codes is introduced.The work of the first and the second authors was partially supported by Spanish grant MTM2011-24858 of the Ministerio de Economía y Competitividad of the Gobierno de España. The work of first author was also partially supported by a grant for research students from the Generalitat Valenciana with reference BFPI/2008/138. The work of the third author was partially supported by the research project UMH-Bancaja with reference IPZS01