A method is described that mosaics linear uniformly redundant arrays and modified uniformly redundant arrays onto a square lattice to create coded aperture patterns of approximately 50% throughput that are contiguous, whereby every opaque element in the aperture is joined to at least one other opaque element. Parameters required to create these perfect high-throughput contiguous apertures for all coded aperture unit pattern orders of up to 300 elements are presented. The apertures presented increase significantly the number of known contiguous apertures in dimensions forbidden to the known patterns and that have the same imaging quality as uniformly redundant arrays and modified uniformly redundant arrays while having imaging quality superior to other known self-supporting apertures. The effect of supporting the weaker areas of these contiguous apertures is investigated. Contiguous apertures are of interest for coded aperture systems that require extra rigidity or use active collimation.