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The weight distribution of a coset of a Reed-Muller code M( 1, m) is invariant under a large transformation group consisting of all affine rearrangements of a vector space with dimension m . We discuss a general algorithm that produces an ordered list of orbit representatives for this group action. As a byproduct the procedure finds the order of the symmetry group of a coset. With m = 6 we can implement the algorithm on a computer and find that there are 150357 equivalence classes. These classes produce 2082 distinct weight distributions. Their symmetry groups have 122 different orders.

A classification of the cosets of the Reed-Muller code $\mathcal{R}(1,6)$