arXiv 2004.008112020-04cs.ITmath.IT

分布式编码的基本限制

Fundamental Limits of Distributed Encoding

Nastaran Abadi Khooshemehr, Mohammad Ali Maddah-Ali

摘要：

在一般的编码理论中，我们经常假设在传输或存储编码符号时会观察到错误，而编码过程本身是无错误的。受最近的应用程序的编码理论，在本文中，我们考虑的情况下，编码的过程是分布式的，容易出错。我们介绍了分布式编码的问题，包括$K\in\mathbb{N}$孤立的源节点和$N\in\mathbb{N}$编码节点。每个源节点具有来自有限域的一个符号，并将其发送到所有编码节点。每个编码节点根据接收到的符号存储编码符号。然而，一些源节点由对手控制，并且可以向不同的编码节点发送不同的符号。根据敌对节点的数量（用$β\in\mathbb{N}$表示）和每个节点生成的符号的数量（用$v\in\mathbb{N}$表示），从编码符号解码的过程可能是不可能的。假设解码器连接到$t \in\mathbb{N}$编码节点的任意子集，并且想要正确地解码诚实节点的符号，而不必识别诚实和对抗节点的集合。本文研究了$t$的最小值$t^*\in\mathbb{N}$，它是$K$，$N$，$β$和$v$的函数.我们表明，当编码节点使用线性编码，$t^*_{\textrm{linear}}=K+2β(v-1)$，如果$N\ge K+2β(v-1)$，和$t^*_{\textrm{linear}}=N$，如果$N\le K+2β(v-1)$。为了实现$t^*_{\textrm{linear}}$，我们使用随机线性编码，并表明，在任何可行的解决方案，解码器发现，诚实的节点的消息被正确解码。相反的基本限制，我们表明，当对手的行为以一种特定的方式，它总是可以混淆解码器之间的两个可行的解决方案，不同的消息至少有一个诚实的节点。

英文摘要：

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 $β\in\mathbb{N}$, 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 $K$, $N$, $β$, and $v$. We show that when the encoding nodes use linear coding, $t^*_{\textrm{linear}}=K+2β(v-1)$, if $N\ge K+2β(v-1)$, and $t^*_{\textrm{linear}}=N$, if $N\le K+2β(v-1)$. 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文