关于有限群中的Ingleton-破坏
On the Ingleton-Violations in Finite Groups
摘要：
给定$n$离散随机变量，其熵向量是由随机变量的所有非空子集的联合熵得到的$2^n-1$维向量。众所周知，在这样的熵向量和从有限群及其子群的$n$得到的某个群可特征化向量之间存在一一对应[3]。这种对应关系对于表征熵向量空间和设计网络码可能是有用的。如果限制注意力的阿贝尔群，那么不是所有的熵向量可以得到。这是对Doughnut等人[4]所示的事实的解释，即线性网络码不能在一般网络编码问题中实现容量。所有阿贝尔群特征向量，以及所有由线性网络码生成的熵向量，都满足一个称为英格尔顿不等式的线性不等式。因此，确定违反英格尔顿不等式的群体是有意义的。在本文中，我们研究的问题，找到非交换有限群，产生特征向量违反英格尔顿不等式。使用改进的计算机搜索，我们发现对称群$S_5$是违反英格尔顿不等式的最小群。仔细研究这个群及其子群的结构，发现它属于Ingleton破坏家族$PGL(2,q)$，具有素数幂$q \geq 5$，即，元素在$\mathbb{F}_q$中的$2\times 2$非奇异矩阵的射影群。我们进一步解释这个家庭使用的理论群体行动。我们还将构造扩展到更一般的群，如$PGL(n,q)$和$GL(n,q)$。因此，这里确定的家庭的群体是很好的候选人，构建网络代码比线性网络代码更强大，我们讨论了一些考虑，构建这样的组网络代码。
英文摘要：
Given $n$ discrete random variables, its entropy vector is the $2^n-1$ dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-to-one correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and $n$ of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al [4] that linear network codes cannot achieve capacity in general network coding problems. All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. It is therefore of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group $S_5$ to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family $PGL(2,q)$ with a prime power $q \geq 5$, i.e., the projective group of $2\times 2$ nonsingular matrices with entries in $\mathbb{F}_q$. We further interpret this family using the theory of group actions. We also extend the construction to more general groups such as $PGL(n,q)$ and $GL(n,q)$. The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes.
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