  We present new constructions of binary quantum codes from quaternary linear
Hermitian self-dual codes. Our main ingredients for these constructions are
nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of
$0$-dimensional binary quantum codes is provided. We give minimum distance
lower bounds for our quantum codes in terms of the minimum distance of their
ingredient linear codes. We also present new results on the minimum distance of
linear cyclic codes using their fixed subcodes. Finally, we list many new
record-breaking quantum codes obtained from our constructions.
