用有限群违反Ingleton不等式
Violating the Ingleton Inequality with Finite Groups
摘要：
众所周知，在n个随机变量集合的熵向量与从有限群及其n个子群获得的某个群可表征向量之间存在一一对应。然而，如果限制注意力的阿贝尔群，那么不是所有的熵向量可以获得。这是对Doughnut等人所示的事实的解释，即线性网络码不能在一般网络编码问题中实现容量（因为线性网络码形成阿贝尔群）。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。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Ingleton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-破坏族PGL（2，p），素数p > 3，即，F_p中元素的2 × 2非奇异矩阵的射影群。因此，这类群是构造比线性网络码更强大的网络码的良好候选者。
英文摘要：
It is well known that there is a one-to-one correspondence between the entropy vector of a collection of n random variables and a certain group-characterizable vector obtained from a finite group and n of its subgroups. However, 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 that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called 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,p) with primes p > 3, i.e., the projective group of 2 by 2 nonsingular matrices with entries in F_p. This family of groups is therefore a good candidate for constructing network codes more powerful than linear network codes.
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