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We present theoretically derived design rules for an absorbing resonance antireflection coating for the spectral range of 100 - 400 nm, applied here on top of a molybdenum-silicon multilayer mirror (Mo/Si MLM) as commonly used in extreme ultraviolet lithography. The design rules for optimal suppression are found to be strongly dependent on the thickness and optical constants of the coating. For wavelengths below λ ∼ 230 nm, absorbing thin films can be used to generate an additional phase shift and complement the propagational phase shift, enabling full suppression already with film thicknesses far below the quarter-wave limit. Above λ ∼ 230 nm, minimal absorption (k < 0.2) is necessary for low reflectance and the minimum required layer thickness increases with increasing wavelength slowly converging towards the quarter-wave limit.As a proof of principle, SixCyNz thin films were deposited that exhibit optical constants close to the design rules for suppression around 285 nm. The thin films were deposited by electron beam co-deposition of silicon and carbon, with N+ ion implantation during growth and analyzed with variable angle spectroscopic ellipsometry to characterize the optical constants. We report a reduction of reflectance at λ = 285 nm, from 58% to 0.3% for a Mo/Si MLM coated with a 20 nm thin film of Si0.52C0.16N0.29.