  We prove new existence and nonexistence results for modular Golomb rulers in
this paper. We completely determine which modular Golomb rulers of order $k$
exist, for all $k\leq 11$, and we present a general existence result that holds
for all $k \geq 3$. We also derive new nonexistence results for infinite
classes of modular Golomb rulers and related structures such as difference
packings, optical orthogonal codes, cyclic Steiner systems and relative
difference families.
