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、分布式计算以及同态秘密共享等应用场景的启发，我们研究了计算编码符号线性组合的通信高效方案。具体而言，设计了低带宽求值方案，用于在给定码字$\pmb{c}\in\mathbb{F}^n$中$d$个剩余分量访问权限的条件下，计算该码字内$\ell$个编码符号的加权和。形式化地，设$\mathbb{F}$是$\mathbb{B}$的$t$次域扩张，$\pmb{c}$是维度为$k$的Reed-Solomon码的码字，我们的任务是计算$\ell$个编码符号的加权和。本文中，对于某些$s<t$，我们提供了一个显式方案，当$d\geq \ell|\mathbb{B}|^s-\ell+k$时，该方案通过从$d$个可用节点下载$d(t-s)$个$\mathbb{B}$中的子符号即可完成此任务。在许多情形下，我们的方案优于现有文献中的方案。此外，我们给出了通用线性码求值方案的特征刻画，并针对Reed-Solomon码的特殊情形，利用该特征刻画推导了求值带宽的下界。