GL(m, 2) acting on R(r, m)/R(r − 1, m) Original Research Article

Let R(r, m) be the rth order Reed-Muller code of length 2m, and let R(r,m)/R(r − 1,m) be the set of all cosets of R(r − 1,m) in R(r,m). The general linear group GL(m,2) acts on R(r,m)/R(r − 1,m). We compute the numbers of the GL(m,2)-orbits of R(r,m)/R(r − 1,m) for 6 ⩽ m ⩽ 11. This is done through a formula for the size of the centralizer of a matrix in GL(m,2) and the observation that any A ∈ GL(m,2) acts on R(r,m)/R(r − 1,m) as a linear isomorphism whose matrix with respect to the basis {Πi∈S Xi + R(r − 1,m) : S ⊂ {1, …, m}, |S| = r} of R(r,m)/R(r − 1, m) is the rth compound of A. We then classify R(3, 6)/R(2, 6) and R(3, 7)/R(2, 7). The implication of these classifications concerning the covering properties of R(2,6) and R(2,7) is also given.