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An array of a large number of LEDs will be widely used in future indoor illumination systems. In this paper, we investigate the problem of rendering uniform illumination by a regular LED array on the ceiling of a room. We first present two general results on the scaling property of the basic illumination pattern, i.e., the light pattern of a single LED, and the setting of LED illumination levels, respectively. Thereafter, we propose to use the relative mean squared error as the cost function to measure the uniformity of the realized illumination pattern, and provide an analysis for this cost function. Based on the analysis, the design of a basic illumination pattern is discussed. The performances of a few basic illumination patterns are compared, and an approach for optimizing the basic illumination pattern through a weighted combination of these basic patterns is also proposed. A weighted combination of Gaussian and raised-cosine functions is found to yield the best results. Finally, three basic regular grid shapes for an LED array are compared. The results show that 13% and 39% of LEDs can be saved for the same degree of uniformity, using the hexagonal instead of the rectangular and triangular grid, respectively.