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

Abstract Let R(r,m) be the r th order Reed-Muller code of length 2 m . 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 Mobius 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.