AGL(M,2) ACTING ON R(R,M) R(S,M)

Let AGL(m, F) be the general affine group of order m over an arbitrary field F. We determine the conjugacy classes of AGL(m, F). When F = GF (q), we also determine the sizes of the centralizers of the elements i n AGL(m, F). The group AGL(m, 2) (= AGL(m, GF(2))) acts on each of the Reed-Muller codes of length 2(m) as an automorphism group. We denote the rth order Reed-Muller code of length 2(m) by R(r, m) and prove tha t under the action of AGL(m,2) the number of orbits of R(t, m)/R(s, m) is equal to that of R(m - (s + 1), m)/R(m - (t + 1), m) for -1 less t han or equal to s < t less than or equal to m. We also compute the num bers of AGL(m, 2)-orbits of R(t, m)/R(s, m) for m = 6, 7, and -1 less than or equal to s < t less than or equal to m. (C) 1995 Academic Pres s. Inc,