  We study lines through the origin of finite-dimensional complex vector spaces
that enjoy a doubly transitive automorphism group. In doing so, we make
fundamental connections with both discrete geometry and algebraic
combinatorics. In particular, we show that doubly transitive lines are
necessarily optimal packings in complex projective space, and we introduce a
fruitful generalization of regular abelian distance-regular antipodal covers of
the complete graph.
