We study the weight distribution of Reed-Muller codes. Prior results
of Kasami and Tokura from the 70's imply bounds on the weight
distribution of the code that apply only until twice the minimum
distance. Since their work there was no progress on this problem.

We use the conflict between structure and randomness to provide
accumulative bounds for the weight distribution of Reed-Muller codes
that are asymptotically tight and apply to *all* distances.

We use similar ideas to study the list-decoding size of Reed-Muller
codes. Given a received word and a distance parameter, we are
interested in bounding the size of the list of Reed-Muller codewords
that are within that distance from the received word. Previous
bounds on the list size of Reed-Muller codes apply only up to the
minimum distance of the code. In this work we provide asymptotic
bounds for the list-decoding size of Reed-Muller codes that apply
for *all* distances.