Motivated by applications in distributed storage, distributed computing, and homomorphic secret sharing, we study communication-efficient schemes for computing linear combinations of coded symbols. Specifically, we design low-bandwidth schemes that evaluate the weighted sum of $\ell$ coded symbols in a codeword $\pmb{c}\in\mathbb{F}^n$, when we are given access to $d$ of the remaining components in $\pmb{c}$. Formally, suppose that $\mathbb{F}$ is a field extension of $\mathbb{B}$ of degree $t$. Let $\pmb{c}$ be a codeword in a Reed-Solomon code of dimension $k$ and our task is to compute the weighted sum of $\ell$ coded symbols. In this paper, for some $s<t$, we provide an explicit scheme that performs this task by downloading $d(t-s)$ sub-symbols in $\mathbb{B}$ from $d$ available nodes, whenever $d\geq \ell|\mathbb{B}|^s-\ell+k$. In many cases, our scheme outperforms previous schemes in the literature. Furthermore, we provide a characterization of evaluation schemes for general linear codes. Then in the special case of Reed-Solomon codes, we use this characterization to derive a lower bound for the evaluation bandwidth.
翻译:受分布式存储、分布式计算和同质秘密共享应用的驱动,我们研究了计算编码符号线性组合的通信效率计划。具体地说,我们设计了低带宽度计划,在代码词$\pmb{c ⁇ in\mathbb{F}$中,当我们获得$\pmb{c}的剩余元件的美元时,我们用分布式存储、分布式计算和同质秘密共享,我们研究的是计算编码符号线性组合的通信效率计划。具体地说,我们设计了低带宽度计划,在标准值代码的Reed-Solomon代码中,我们设计了美元和美元代码的加权总和。对于一些美元来说,我们提供了一个明确的计划,通过下载$(t-s-s)$(mathb{B}的域值$($mathb{B}$$)。让我们用$\pmb{c}美元作为代码的编码的代码,每当我们的特殊- Sload\\\max case a case, 只要我们的特殊- crudeal a dedeal a.