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、分布式计算和同态秘密共享等应用的驱动，我们研究了用于计算编码符号线性组合的通信高效方案。具体而言，我们设计了低带宽方案，在可访问码字$\pmb{c}\in\mathbb{F}^n$中$d$个其他分量时，评估$\ell$个编码符号的加权和。形式上，假设$\mathbb{F}$是$\mathbb{B}$的$t$次域扩张，设$\pmb{c}$为维度为$k$的里德-所罗门码中的码字，需计算$\ell$个编码符号的加权和。本文针对$s<t$的情形，提供了一种显式方案：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$\mathbb{B}$中的子符号完成任务。在许多情况下，本方案优于文献中的已有方案。此外，我们给出了通用线性码评估方案的特征刻画，并在里德-所罗门码的特殊情形下利用该特征推导了评估带宽的下界。