We present the first 7-millimetre observations of two high-redshift, Lya-bright radio galaxies (MRC 2104-242 and MRC 0943-242) performed with the 2 x 2 GHz instantaneous bandwidth of the Compact Array Broadband Backend (CABB) at the Australia Telescope Compact Array (ATCA). The aim was to search for 12CO(1-0) emission in these systems and test the millimetre capabilities of CABB for performing spectral line observations at high redshifts. We show that the stable band and enhanced velocity coverage of CABB, combined with hybrid array configurations, provide the ATCA with excellent 7-mm capabilities that allow reliable searches for the ground transition of CO at high redshifts. In this paper we explicitly discuss the calibration procedures used to reach our results. We set a firm upper limit to the mass of molecular gas in MRC 2104-242 (z = 2.5) of M(H2) < 2 x 10^10 (alpha_x/0.8) M_sun. For MRC 0943-242 (z=2.9) we derive an upper limit of M(H2) < 6 x 10^10 (alpha_x/0.8) M_sun. We also find a tentative 3-sigma CO detection in the outer part of the giant Lya halo that surrounds MRC 0943-242. The 30-33 GHz radio continuum of MRC 2104-242 and MRC 0943-242 is reliably detected. Both radio sources show a spectral index of alpha ~ -1.5 between 1.4 and 30 GHz, with no evidence for spectral curvature within this range of frequencies.
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