paper

Coxeter codes: Extending the Reed-Muller family

arXiv:2502.14746 · doi:10.1007/s10623-025-01749-y

Abstract

Binary Reed-Muller (RM) codes are defined via evaluations of Boolean-valued functions on $Z_{2}^{m}$ . We introduce a class of binary linear codes that generalizes the RM family by replacing the domain $Z_{2}^{m}$ 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.

v1: Extended abstract, v2: full paper, v3 has added new material on quantum Coxeter code and a remark on decoding Coxeter codes. This version is the final form of the paper, published in Designs, Codes and Cryptography