Constructions of Batch Codes via Finite Geometry

A primitive $k$-batch code encodes a string $x$ of length $n$ into string $y$ of length $N$, such that each multiset of $k$ symbols from $x$ has $k$ mutually disjoint recovering sets from $y$. We develop new explicit and random coding constructions of linear primitive batch codes based on finite geometry. In some parameter regimes, our proposed codes have lower redundancy than previously known batch codes.Comment: 7 pages, 1 figure, 1 tabl

Bounds and Constructions for Generalized Batch Codes

Private information retrieval (PIR) codes and batch codes are two important types of codes that are designed for coded distributed storage systems and private information retrieval protocols. These codes have been the focus of much attention in recent years, as they enable efficient and secure storage and retrieval of data in distributed systems. In this paper, we introduce a new class of codes called \emph{$(s,t)$-batch codes}. These codes are a type of storage codes that can handle any multi-set of $t$ requests, comprised of $s$ distinct information symbols. Importantly, PIR codes and batch codes are special cases of $(s,t)$-batch codes. The main goal of this paper is to explore the relationship between the number of redundancy symbols and the $(s,t)$-batch code property. Specifically, we establish a lower bound on the number of redundancy symbols required and present several constructions of $(s,t)$-batch codes. Furthermore, we extend this property to the case where each request is a linear combination of information symbols, which we refer to as \emph{functional $(s,t)$-batch codes}. Specifically, we demonstrate that simplex codes are asymptotically optimal functional $(s,t)$-batch codes, in terms of the number of redundancy symbols required, under certain parameter regime.Comment: 25 page

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