Function-Correcting Codes

Lenz, Andreas; Bitar, Rawad; Wachter-Zeh, Antonia; Yaakobi, Eitan

doi:10.1109/isit45174.2021.9517976

“…In our work, we focus on this parameter regime, that is, t is small.• Low-bandwidth function evaluation of Reed-Solomon encoded data. Recently, Lenz et alinvestigated the problem of designing codes for function evaluation of information symbols transmitted over a noisy channel[LBWZY21]. In the paper, the authors analyzed the trade-off between channel noise and coding rates for general coding channels.…”

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confidence: 99%

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

Kiah,

Kim,

Kruglik

et al. 2024

IEEE Trans. Inform. Theory

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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 coded symbols in a codeword c c c ∈ F n , when we are given access to d of the remaining components in c c c.Formally, suppose that F is a field extension of B of degree t. Let c c c be a codeword in a Reed-Solomon code of dimension k and our task is to compute the weighted sum of 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 B from d available nodes, whenever d ≥ |B| s − + 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.

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

Kiah,

Kim,

Kruglik

et al. 2023

2023 IEEE International Symposium on Information Theory (ISIT)

4

0

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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 coded symbols in a codeword c c c ∈ F n , when we are given access to d of the remaining components in c c c.Formally, suppose that F is a field extension of B of degree t. Let c c c be a codeword in a Reed-Solomon code of dimension k and our task is to compute the weighted sum of 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 B from d available nodes, whenever d ≥ |B| s − + 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.

show abstract