Motivated by the use of trace codes in low-bandwidth repair, we investigate the decoding of a Reed-Solomon subcode whose information polynomials are characterized by a known sparse support. We ask: Can we correct more errors by leveraging the known structure of the information polynomial? We affirmatively respond to this query by introducing two schemes designed to complement any Reed-Solomon decoder. Our findings demonstrate better error-correcting capabilities than the naive way to decode based on the maximum degree of the information polynomial.

We consider the problem of evaluating distinct multivariate polynomials over several massive datasets in a distributed computing system with a single master node and multiple worker nodes. We focus on the general case when each multivariate polynomial is evaluated over its corresponding dataset and propose a generalization of the Lagrange Coded Computing framework (Yu et al., 2019) to perform all computations simultaneously while providing robustness against stragglers who do not respond in time, adversarial workers who respond with wrong computation and information-theoretic security of dataset against colluding workers. Our scheme introduces a small computation overhead which results in a reduction in download cost and also offers comparable resistance to stragglers over existing solutions. On top of it, we also propose two verification schemes to detect the presence of adversaries, which leads to incorrect results, without involving additional nodes.

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 ℓ coded symbols in a codewordc∈ Fn, when we are given access todof the remaining components inc. Formally, suppose that F is a field extension of B of degreet. Letcbe a codeword in a Reed-Solomon code of dimensionkand our task is to compute the weighted sum of ℓ coded symbols. In this paper, for somest, we provide an explicit scheme that performs this task by downloadingd(t-s) sub-symbols in B fromdavailable nodes, wheneverd≥ ℓ|B|s-ℓ +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.

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 ℓ coded symbols in a codeword ${\mathbf{c}} \in {\mathbb{F}^n}$, when we are given access to d of the remaining components in c. Formally, suppose that $\mathbb{F}$ is a field extension of $\mathbb{B}$ of degree t. Let c be a codeword in a Reed-Solomon code of dimension k and our task is to compute the weighted sum of ℓ 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.

We consider the repair scheme of Guruswami-Wootters for the Reed-Solomon code and ask: can we correctly repair a failed node in the presence of erroneous nodes? Equivalently, we consider the collection of downloaded traces as a code and investigate its code-distance properties. We propose three lower bounds on its minimum distance and study methods to efficiently correct errors close to these bounds.

We consider the problem of evaluating arbitrary multivariate polynomials over several massive datasets in a distributed computing system with a single master node and multiple worker nodes. We focus on the general case when each multivariate polynomial is evaluated over its dataset and propose a generalization of the Lagrange Coded Computing framework (Yu et al. 2019) to provide robustness against stragglers who do not respond in time, adversarial workers who respond with wrong computation and information-theoretic security of dataset against colluding workers. Our scheme introduces a small computation overhead which results in a reduction in download cost and also offers comparable resistance to stragglers over existing solutions.