  We report on coherent propagation of antiferromagnetic (AFM) spin waves over
a long distance ($\sim$10 $\mu$m) at room temperature in a canted AFM
$\alpha$-Fe$_2$O$_3$ with the Dzyaloshinskii-Moriya interaction (DMI).
Unprecedented high group velocities (up to 22.5 km/s) are characterized by
microwave transmission using all-electrical spin wave spectroscopy. We derive
analytically AFM spin-wave dispersion in the presence of the DMI which accounts
for our experimental results. The AFM spin waves excited by nanometric coplanar
waveguides with large wavevectors enter the exchange regime and follow a
quasi-linear dispersion relation. Fitting of experimental data with our
theoretical model yields an AFM exchange stiffness length of 1.7 angstrom. Our
results provide key insights on AFM spin dynamics and demonstrate high-speed
functionality for AFM magnonics.
