On Rank Problems for Subspaces of Matrices over Finite Fields

In this thesis we are concerned with themes suggested by rank properties of subspaces of matrices. Historically, most work on these topics has been devoted to matrices over such fields as the real or complex numbers, where geometric or analytic methods may be applied. Such techniques are not obviously applicable to finite fields, and there were very few general theorems relating to rank problems over finite fields. In this thesis we are concerned mainly with constant rank subspaces of matrices over finite fields, with particular focus on two subcases: (1) constant rank subspaces of symmetric or hermitian matrices; and (2) constant full rank subspaces of matrices, which correspond to nonassociative algebraic structures known as semifields. In Chapter 1 we will introduce constant rank subspaces of matrices, and review the known results on the maximum dimension of such a subspace. In Chapter 2 we will recall the definition of a semifield, and illustrate how these algebraic structures are related to constant rank subspaces of matrices. In Chapter 3 we will prove a general theorem on subspaces of function spaces, and apply the results to obtain new upper bounds on subspaces of matrices, which are sharp in some cases. In Chapter 4 we will study primitive elements in finite semifields, and prove their existence for a certain family of semifields. In Chapters 5 and 6 we will introduce a construction for semifields using skew-polynomial rings. We will show how they are related to other known constructions, use this representation to obtain new results, and provide elegant new proofs for some known results.