The eigenmatrix of the linear association scheme on \(R(2,m)\) (Q5939923)

Let \({\mathcal P}_m\) be the \(F_2\)-algebra of all functions from \(F_2^m\) to \(F_2\): NEWLINE\[NEWLINE{\mathcal P}_m=F_2[X_1,\dots ,X_m]/(X_1^2-X_1,\dots ,X_m^2-X_m).NEWLINE\]NEWLINE The Hamming weight of an \(f\in {\mathcal P}_m\) is \(|f|=|f^{-1}(1)|\). For each \(r\leq m\), \(R(r,m)=\{f\in {\mathcal P}_m\mid \deg f\leq r\}\) is the \(r\)th order Reed-Muller code of length \(2^m\). For \(-1\leq r\leq s\leq m\), the action of the general affine group \(\text{AGL}(m,2)\) on \(R(s,m)/R(r,m)\) defines a linear association scheme. Let the scheme on \(R(s,m)\) be the linear association scheme \(R(s,m)/R(-1,m)\). The eigenmatrix of the association scheme of symplectic forms \(R(2,m)/R(1,m)\) was determined by Delsarte and Goethals. In this paper the eigenmatrix of the linear association scheme on \(R(2,m)\) is determined. As a consequence, explicit formulas for the weight enumerators of all cosets of \(R(m-3,m)\) in \(R(m-2,m)\) are obtained.