Abstract
We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the ‘essence of Reed-Mullerity’. The idea is to find a ‘seed’ upper bound—a properly chosen combination of binomial coefficients—well fitted to the respective growths of m (log of length) and r (order), to initiate double induction on m and r. Suprisingly enough, these two simple ingredients suffice to essentially fill the gaps between lower and upper bounds, a result stated in our theorem.
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