Coxeter codes: Extending the Reed-Muller family

Binary Reed-Muller (RM) codes are defined via evaluations of Boolean-valued functions on ${\mathbb{Z}}_2^m$. We introduce a class of binary linear codes that generalizes the RM family by replacing the domain ${\mathbb{Z}}_2^m$ 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.