We have used the SEST 15-metre and Onsala 20-metre telescopes to perform deep (r.m.s. >~ 30 mJy) integrations of various molecular rotational transitions towards damped Lyman-alpha absorption systems (DLAs) known to occult millimetre-loud quasars. We have observed 6 new systems and improved the existing limits for 11 transitions. These limits may be approaching the sensitivities required to detect new systems and we present a small number of candidate systems which we believe warrant further observation.
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