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, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprising of $K\in\mathbb{N}$ isolated source nodes and $N\in\mathbb{N}$ encoding nodes. Each source node has one symbol from a finite field and sends it to all encoding nodes. Each encoding node stores an encoded symbol, 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 $\beta\in\mathbb{N}$, and the number of symbols that each one generates, denoted by $v\in\mathbb{N}$, the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in\mathbb{N}$ 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 study $t^*\in\mathbb{N}$, the minimum of $t$, which is a function of $K$, $N$, $\beta$, and $v$. We show that when the encoding nodes use linear coding, $t^*_{\textrm{linear}}=K+2\beta(v-1)$, if $N\ge K+2\beta(v-1)$, and $t^*_{\textrm{linear}}=N$, if $N\le K+2\beta(v-1)$. In order to achieve $t^*_{\textrm{linear}}$, we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. For the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node.