在有噪信道上的无差错传输中的用户识别
Privileged users in zero-error transmission over a noisy channel
摘要：
图$G$的$k$ -次幂是其顶点集为$V(G)^k$的图，其中两个不同的$k$ -元组相邻当且仅当它们在$G$中的每个坐标上相等或相邻。$G$的香农容量$c(G)$是$\lim_{k\to\infty}α(G^k)^{1/k}$，其中$α(G)$表示$G$的独立数。当$G$是信道$\mathcal{C}$的特征图时，$c(G)$测量零错误协议中$\mathcal{C}$的有效字母大小。通道之和$\mathcal{C}=\sum_i \mathcal{C}_i$描述了当存在$t\geq 2$时的设置，每个$t\geq 2$都有自己的通道$\mathcal{C}_i$，并且单词中的每个字母都可以从任何一个通道中选择。这对应于特征图$G=\sum_i G_i$的不相交并集。我们发现，对于任何固定的$t$和任何家庭$F$的子集的$T={1,2,...,t}$，有$t$图$G_1,G_2, ...,G_t$，使每个子集$I$的$T$，香农容量的不相交的联盟$\sum_{i \in I} G_i$是“大”，如果$I$包含一个成员的$F$，否则是“小”。
英文摘要：
The $k$-th power of a graph $G$ is the graph whose vertex set is $V(G)^k$, where two distinct $k$-tuples are adjacent iff they are equal or adjacent in $G$ in each coordinate. The Shannon capacity of $G$, $c(G)$, is $\lim_{k\to\infty}α(G^k)^{1/k}$, where $α(G)$ denotes the independence number of $G$. When $G$ is the characteristic graph of a channel $\mathcal{C}$, $c(G)$ measures the effective alphabet size of $\mathcal{C}$ in a zero-error protocol. A sum of channels, $\mathcal{C}=\sum_i \mathcal{C}_i$, describes a setting when there are $t\geq 2$ senders, each with his own channel $\mathcal{C}_i$, and each letter in a word can be selected from either of the channels. This corresponds to a disjoint union of the characteristic graphs, $G=\sum_i G_i$. We show that for any fixed $t$ and any family $F$ of subsets of $T={1,2,...,t}$, there are $t$ graphs $G_1,G_2, ...,G_t$, so that for every subset $I$ of $T$, the Shannon capacity of the disjoint union $\sum_{i \in I} G_i$ is "large" if $I$ contains a member of $F$, and is "small" otherwise.
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。
