Coxeter codes: Extending the Reed-Muller family
Abstract
Binary Reed-Muller (RM) codes are defined via evaluations of Boolean-valued functions on . We introduce a class of binary linear codes that generalizes the RM family by replacing the domain 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 rotations can perform non-trivial logic.
Keywords
Cite
@article{arxiv.2502.14746,
title = {Coxeter codes: Extending the Reed-Muller family},
author = {Nolan J. Coble and Alexander Barg},
journal= {arXiv preprint arXiv:2502.14746},
year = {2026}
}
Comments
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