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 $(M,R) = (1/2,5/3)$.

翻译：本文第一部分发展了线性代数中的若干定理，可应用于当所有随机变量均为独立比特源中各比特的线性函数时的信息论问题。我们称向量空间的一组子空间集合是"协调的"，若该向量空间存在一组基，使得每个子空间均由该基与其交集中的元素张成。我们通过一个称为该族子空间"失调量"的不变量来度量子空间集合未能协调的程度，并建立了关于失调量的基础性结果。特别地，这些结果给出了涉及向量空间中三个子空间的若干新公式。随后，我们应用部分结果及Tian的方法，在基本编码缓存问题的特例中获得了新的下界。使用该问题的常规符号，我们证明：在$N=3$个文档与$K=3$个缓存的条件下，对于实现存储-速率对$(M,R)$的线性方案，有$6M+5R\ge 11$。同时给出$N=K=3$情形下实现存储-速率对$(M,R) = (1/2,5/3)$的新缓存方案。