Let $G$ be a connected graph on $n$ vertices and $C$ be an $(n,k,d)$ code with $d\ge 2$, defined on the alphabet set $\{0,1\}^m$. Suppose that for $1\le i\le n$, the $i$-th vertex of $G$ holds an input symbol $x_i\in\{0,1\}^m$ and let $\vec{x}=(x_1,\ldots,x_n)\in\{0,1\}^{mn}$ be the input vector formed by those symbols. Assume that each vertex of $G$ can communicate with its neighbors by transmitting messages along the edges, and these vertices must decide deterministically, according to a predetermined communication protocol, that whether $\vec{x}\in C$. Then what is the minimum communication cost to solve this problem? Moreover, if $\vec{x}\not\in C$, say, there is less than $\lfloor(d-1)/2\rfloor$ input errors among the $x_i$'s, then what is the minimum communication cost for error correction? In this paper we initiate the study of the two problems mentioned above. For the error detection problem, we obtain two lower bounds on the communication cost as functions of $n,k,d,m$, and our bounds are tight for several graphs and codes. For the error correction problem, we design a protocol which can efficiently correct a single input error when $G$ is a cycle and $C$ is a repetition code. We also present several interesting problems for further research.

中文翻译：

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通信网络中的错误检测和纠正

令 $G$ 是 $n$ 个顶点上的连通图，$C$ 是 $(n,k,d)$ 代码，$d\ge 2$，在字母集 $\{0,1\} 上定义^米$。假设对于 $1\le i\le n$，$G$ 的第 $i$ 个顶点持有一个输入符号 $x_i\in\{0,1\}^m$ 并且让 $\vec{x}= (x_1,\ldots,x_n)\in\{0,1\}^{mn}$ 是由这些符号形成的输入向量。假设$G$的每个顶点都可以通过沿边传输消息与其邻居进行通信，并且这些顶点必须根据预定的通信协议确定性地决定$\vec{x}\in C$。那么解决这个问题的最小通信成本是多少呢？此外，如果 $\vec{x}\not\in C$，比如说，在 $x_i$ 中输入错误少于 $\lfloor(d-1)/2\rfloor$，那么纠错的最小通信成本是多少？在本文中，我们开始研究上述两个问题。对于错误检测问题，我们获得了作为 $n,k,d,m$ 函数的通信成本的两个下限，并且我们的边界对于几个图和代码是严格的。对于纠错问题，我们设计了一个协议，当 $G$ 是一个循环，$C$ 是一个重复码时，它可以有效地纠正单个输入错误。我们还提出了几个有趣的问题以供进一步研究。我们设计了一个协议，当 $G$ 是一个循环并且 $C$ 是一个重复代码时，它可以有效地纠正单个输入错误。我们还提出了几个有趣的问题以供进一步研究。我们设计了一个协议，当 $G$ 是一个循环并且 $C$ 是一个重复代码时，它可以有效地纠正单个输入错误。我们还提出了几个有趣的问题以供进一步研究。