  We present Lyman continuum (LyC) radiation escape fraction $f_{\rm{esc}}$
measurements for 183 spectroscopically confirmed star-forming galaxies in the
redshift range $3.11 < z < 3.53$ in the \textit{Chandra} Deep Field South. We
use ground-based imaging to measure $f_{\rm{esc}}$, and use ground- and
space-based photometry to derive galaxy physical properties using spectral
energy distribution (SED) fitting. We additionally derive [O III]+H$\beta$
equivalent widths (that fall in the observed K band) by including nebular
emission in the SED fitting. After removing foreground contaminants, we report
the discovery of 11 new candidate LyC leakers, with absolute LyC escape
fractions, $f_{\rm{esc}}$ in the range $0.14-0.85$. From non-detections, we
place $1\sigma$ upper limits of $f_{\rm{esc}}<0.12$, where the Lyman-break
selected galaxies have $f_{\rm{esc}} < 0.11$ and `blindly' discovered galaxies
with no prior photometric selection have $f_{\rm{esc}}<0.13$. We find a
slightly higher $1\sigma$ limit of $f_{\rm{esc}}<0.20$ for extreme emission
line galaxies with rest-frame [O III]+H$\beta$ equivalent widths $>300$A. For
candidate LyC leakers, we find a weak negative correlation between
$f_{\rm{esc}}$ and galaxy stellar masses, no correlation between $f_{\rm{esc}}$
specific star-formation rates (sSFRs) and a positive correlation between
$f_{\rm{esc}}$ and EW$_0$([O III]+H$\beta$). The weak/no correlations between
stellar mass and sSFRs may be explained by misaligned viewing angles and/or
non-coincident timescales of starburst activity and periods of high
$f_{\rm{esc}}$. Alternatively, escaping radiation may predominantly occur in
highly localised star-forming regions, or $f_{\rm{esc}}$ measurements may be
impacted by stochasticity of the intervening neutral medium, obscuring any
global trends with galaxy properties. These hypotheses have important
consequences for models of reionisation.
