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,textsc{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 $0.07-0.52$. Most galaxies in our sample ($approx94%$) do not show any LyC leakage, and we place $1sigma$ upper limits of $f_{rm{esc}} < 0.07$ through weighted averaging, where the Lyman-break selected galaxies have $f_{rm{esc}} < 0.07$ and `blindly discovered galaxies with no prior photometric selection have $f_{rm{esc}} < 0.10$. We additionally measure $f_{rm{esc}} < 0.09$ for extreme emission line galaxies in our sample with rest-frame [O,textsc{iii}],+,H$beta$ equivalent widths $>300$,AA. For the candidate LyC leakers, we do not find a strong dependence of $f_{rm{esc}}$ on their stellar masses and/or specific star-formation rates, and no correlation between $f_{rm{esc}}$ and EW$_0$([O,textsc{iii}],+,H$beta$). We suggest that this lack of correlations may be explained by viewing angle 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, thereby obscuring any global trends with galaxy properties. Both hypotheses have important consequences for models of reionisation.
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