The combinatorics of binary matrices

This thesis studies combinatorial properties of binary matrices that occur frequently in mathematics and computer science. We show that almost all binary matrices have the property that their structure cannot be inferred from a small subset of their entries. We also study sets of binary matrices that satisfy an important property called orthogonality. We find new tools for diagnosing when a set of orthogonal binary matrices cannot be extended to a larger such set. Sets of orthogonal binary matrices have important applications in areas such as experimental design, coding theory and cryptography.