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We present Lyman continuum (LyC) radiation escape fraction $f_{\rm{esc}}$measurements for 183 spectroscopically confirmed star-forming galaxies in theredshift range $3.11 < z < 3.53$ in the \textit{Chandra} Deep Field South. Weuse ground-based imaging to measure $f_{\rm{esc}}$, and use ground- andspace-based photometry to derive galaxy physical properties using spectralenergy distribution (SED) fitting. We additionally derive [O III]+H$\beta$equivalent widths (that fall in the observed K band) by including nebularemission in the SED fitting. After removing foreground contaminants, we reportthe discovery of 11 new candidate LyC leakers, with absolute LyC escapefractions, $f_{\rm{esc}}$ in the range $0.14-0.85$. From non-detections, weplace $1\sigma$ upper limits of $f_{\rm{esc}}<0.12$, where the Lyman-breakselected galaxies have $f_{\rm{esc}} < 0.11$ and `blindly' discovered galaxieswith no prior photometric selection have $f_{\rm{esc}}<0.13$. We find aslightly higher $1\sigma$ limit of $f_{\rm{esc}}<0.20$ for extreme emissionline galaxies with rest-frame [O III]+H$\beta$ equivalent widths $>300$A. Forcandidate 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 betweenstellar mass and sSFRs may be explained by misaligned viewing angles and/ornon-coincident timescales of starburst activity and periods of high$f_{\rm{esc}}$. Alternatively, escaping radiation may predominantly occur inhighly localised star-forming regions, or $f_{\rm{esc}}$ measurements may beimpacted by stochasticity of the intervening neutral medium, obscuring anyglobal trends with galaxy properties. These hypotheses have importantconsequences for models of reionisation.

No strong dependence of Lyman continuum leakage on physical properties of star-forming galaxies at $\mathbf{3.1 \lesssim z \lesssim 3.5}$