  This work investigates the use of a well-known empirical correlation between
the velocity dispersion, metallicity, and luminosity in H beta of nearby HII
galaxies to measure the distances to HII-like starburst galaxies at high
redshifts. This correlation is applied to a sample of 15 starburst galaxies
with redshifts between z=2.17 and z=3.39 to constrain Omega_m, using data
available from the literature. A best-fit value of Omega_m = 0.21 +0.30 -0.12
in a Lambda-dominated universe and of Omega_m = 0.11 +0.37 -0.19 in an open
universe is obtained. A detailed analysis of systematic errors, their causes,
and their effects on the values derived for the distance moduli and Omega_m is
carried out. A discussion of how future work will improve constraints on
Omega_m by reducing the errors is also presented.
