Explicit Low-Bandwidth Evaluation Schemes for Weighted Sums of Reed-Solomon-Coded Symbols

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
$s\lt t$
, we provide an explicit scheme that performs this task by downloading
$d(t-s)$
sub-symbols in
$\mathbb {B}$
from d available nodes, whenever
$d\ge \ell |\mathbb {B}|^{s}-\ell +k$
. 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.