Coxeter codes: extending the Reed–Muller family

Abstract

Binary Reed–Muller (RM) codes are defined via evaluations of Boolean-valued functions on Z2m. We introduce a class of binary linear codes that generalizes the RM family by replacing the domain Z2m with an arbitrary finite Coxeter group. Like RM codes, this class is closed under duality, forms a nested code sequence, satisfies a multiplication property, and has asymptotic rate determined by a Gaussian distribution. Coxeter codes also give rise to a family of quantum codes for which transversal diagonal Z rotations can perform non-trivial logic.