The eigenmatrix of the linear association scheme on R(2,m)

Let R(r,m) be the rth order Reed-Muller code of length 2m. For −1⩽r⩽s⩽m, the action of the general affine group AGL(m,2) on R(s,m)/R(r,m) defines a linear association scheme on R(s,m)/R(r,m). In this paper, we determine the eigenmatrix of the linear association scheme on R(2,m) (=R(2,m)/R(−1,m)). Our approach relies on the Möbius inversion and detailed calculations with the general linear group and the symplectic group over GF(2). As a consequence, we obtain explicit formulas for the weight enumerators of all cosets of R(m−3,m). Such explicit formulas were not available previously.