# Fundamental Limits of Distributed Encoding

In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, we introduce the problem of distributed encoding which comprises of a set of K ∈N isolated source nodes and N ∈N encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of adversarial nodes, denoted by β∈N, and the cardinality of the set of symbols that each one generates, denoted by v ∈N, this would make the process of decoding from the encoded symbols impossible. Assume that a decoder connects to an arbitrary subset of t ∈N encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. In this paper, we characterize t^* ∈N, as the minimum of such t, as a function of K, N, β, and v. We show that for β≥ 1, v> 2, t^*=K+β (v-1)+1, if N ≥ K+β (v-1)+1, and t^*=N, if N < K+β (v-1). In order to achieve t^*, we introduce a nonlinear code. In continue, we focus on linear coding and show that t^*_linear=K+2β(v-1), if N> K+2β(v-1), and t^*_linear=N, if N< K+2β(v-1).

## Authors

• 2 publications
• 28 publications
• ### Real-time oblivious erasure correction with linear time decoding and constant feedback

We continue the study of rateless codes for transmission of information ...
01/26/2021 ∙ by Shashwat Silas, et al. ∙ 0

• ### Treeplication: An Erasure Code for Distributed Full Recovery under the Random Multiset Channel

This paper presents a new erasure code called Treeplication designed for...
02/23/2020 ∙ by Michael Gandelman, et al. ∙ 0

• ### Timely Lossless Source Coding for Randomly Arriving Symbols

We consider a real-time streaming source coding system in which an encod...
10/02/2018 ∙ by Jing Zhong, et al. ∙ 0

• ### Property Invariant Embedding for Automated Reasoning

Automated reasoning and theorem proving have recently become major chall...
11/27/2019 ∙ by Miroslav Olšák, et al. ∙ 0

• ### Power packet transferability via symbol propagation matrix

Power packet is a unit of electric power transferred by a power pulse wi...
08/08/2017 ∙ by Shinya Nawata, et al. ∙ 0

• ### Efficient Trimmed Convolutional Arithmetic Encoding for Lossless Image Compression

Arithmetic encoding is an essential class of coding techniques which hav...
01/15/2018 ∙ by Mu Li, et al. ∙ 0