Orthogonal Polynomials and the MacWilliams Transform for Permutation-Invariant Qudit Codes

We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes.