  Star-forming galaxies are the sources likely to have reionized the universe.
As we cannot observe them directly due to the opacity of the intergalactic
medium at $z\gtrsim5$, we study $z\sim3\text{--}5$ galaxies as proxies to place
observational constraints on cosmic reionization. Using new deep \textit{Hubble
Space Telescope} rest-frame UV F336W and F435W imaging (30-orbit,
$\sim40$~arcmin$^2$, $\sim29\text{--}30$~mag depth at 5$\sigma$), we attempt to
identify a sample of Lyman continuum galaxies (LCGs). These are individual
sources that emit ionizing flux below the Lyman break ($<912~\text{\AA}$). This
population would allow us to constrain cosmic reionization parameters such as
the number density and escape fraction ($f_{\rm esc}$) of ionizing sources. We
compile a comprehensive parent sample that does not rely on the Lyman-break
technique for redshifts. We present three new spectroscopic candidates at
$z\sim3.7\text{--}4.4$, and 32 new photometric candidates. The high-resolution
multi-band HST imaging and new Keck/Low Resolution Imaging Spectrometer (LRIS)
redshifts make these promising spectroscopic LCG candidates. Using both a
traditional and probabilistic approach, we find the most likely $f_{\rm esc}$
values for the three spectroscopic LCG candidates are $>100\%$, and therefore
not physical. We are unable to confirm the true nature of these sources with
the best available imaging and direct blue Keck/LRIS spectroscopy. More
spectra, especially from the new class of 30 m telescopes, will be required to
build a statistical sample of LCGs to place firm observational constraints on
cosmic reionization.
