Improving the upper bounds on the covering radii of Reed-Muller codes

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami and Tokura, of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders