Coordination and Discoordination in Linear Algebra, Linear Information Theory, and Coded Caching

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