论文标题
线性代数，线性信息理论和编码缓存中的协调和脱节
Coordination and Discoordination in Linear Algebra, Linear Information Theory, and Coded Caching
论文作者
论文摘要
在本文的第一部分中，当涉及的所有随机变量都是独立位源的单个位的线性函数时，我们将在线性代数中开发一些定理。 我们说，如果矢量空间的基础使每个子空间与基础相交跨越，则矢量空间的子空间的集合被“协调”。我们衡量了一个不变的子空间集合的失败，我们称之为家庭的“脱节”。我们开发了有关脱节的一些基本结果。特别是，这些结果给出了许多涉及向量空间的三个子空间的新公式。 然后，我们应用了许多结果，以及Tian的方法，以在基本编码的缓存问题的特殊情况下获得一些新的下限。就这些问题的常用符号而言，我们表明，对于$ n = 3 $文档和$ k = 3 $缓存，假设该方案是线性的，那么我们有600万美元+5r \ ge 11 $，对于实现内存率对$（m，r）$的方案。我们还为$ n = k = 3 $提供了一个新的缓存方案，该方案可实现$（m，r）=（1/2,5/3）$。
In the first part of this paper we develop some theorems in linear algebra applicable to information theory when all random variables involved are linear functions of the individual bits of a source of independent bits. We say that a collection of subspaces of a vector space are "coordinated" if the vector space has a basis such that each subspace is spanned by its intersection with the basis. We measure the failure of a collection of subspaces to be coordinated by an invariant that we call the "discoordination" of the family. We develop some foundational results regarding discoordination. In particular, these results give a number of new formulas involving three subspaces of a vector space. We then apply a number of our results, along with a method of Tian to obtain some new lower bounds in a special case of the basic coded caching problem. In terms of the usual notation for these problems, we show that for $N=3$ documents and $K=3$ caches, we have $6M+5R\ge 11$ for a scheme that achieves the memory-rate pair $(M,R)$, assuming the scheme is linear. We also give a new caching scheme for $N=K=3$ that achieves the pair $(M,R) = (1/2,5/3)$.
