We present Lyman continuum (LyC) radiation escape fraction (f_esc) measurements for 183 spectroscopically confirmed star-forming galaxies in the redshift range 3.11 < z < 3.53 in the Chandra Deep Field South. We use ground-based imaging to measure f_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 β equivalent widths (that fall in the observed K band) by including nebular emission in SED fitting. After removing foreground contaminants, we report the discovery of 11 new candidate LyC leakers at ≳ 2σ level, with f_esc in the range 0.14−0.85. From non-detections, we place 1σ upper limits of f_esc <0.12, where the Lyman-break selected galaxies have f_esc <0.11 and ‘blindly’ discovered galaxies with no prior photometric selection have f_esc <0.13. We find a slightly higher 1σ limit of f_esc <0.20 from extreme emission line galaxies with rest-frame O iii  +  H β equivalent widths >300 Å. For candidate LyC leakers, we find a weak negative correlation between f_esc and galaxy stellar masses, no correlation between f_esc and specific star-formation rates (sSFRs) and a positive correlation between f_esc and EW_0(O iii  +  H β). The weak/no correlations between stellar mass and sSFRs may be explained by misaligned viewing angles and/or non-coincident time-scales of starburst activity and periods of high f_esc. Alternatively, escaping radiation may predominantly occur in highly localized star-forming regions, or f_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 reionization.