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The CORE collaboration thanks CNES, Thales Alenia Space, and Air Liquide Advanced Technologies for advice and technical support during the preparation of the CORE proposal. We also thank the ESA CDF team for the CMB Polarisation CDF study performed in March 2016, the results of which were extensively used to define the mission concept presented in this paper. J.G.N. acknowledges financial support from the Spanish MINECO for a Ramon y Cajal fellowship (RYC-2013-13256) and the I+D 2015 project AYA2015-65887-P (MINECO/FEDER). CJM is supported by an FCT Research Professorship, contract reference IF/00064/2012, funded by FCT/MCTES (Portugal) and POPH/FSE. F.J.C., R.F.-C., E.M.-G. and P.V. acknowledge support from the Spanish Ministerio de Econom´ıa y Competitividad project ESP2015-70646-C2-1-R (cofinanced with EU FEDER funds), ConsoliderIngenio 2010 project CSD2010-00064 and from the CSIC “Proyecto Intramural Especial” project 201550E091. FA is supported by the National Taiwan University (NTU) under Project No. 103R4000 and by the NTU Leung Center for Cosmology and Particle Astrophysics (LeCosPA) under Project No. FI121. BFR acknowledges support from the National Science Centre, Poland, under grant 2014/13/B/ST9/00845.