Coxeter codes: Extending the Reed-Muller family

TL;DR

This paper introduces Coxeter codes, a generalization of Reed-Muller codes based on finite Coxeter groups, which retain key properties and enable new quantum error correction techniques.

Contribution

It extends Reed-Muller codes to arbitrary Coxeter groups, preserving important algebraic properties and enabling quantum code applications.

Findings

Coxeter codes are closed under duality.

They form nested code sequences.

They enable transversal diagonal Z rotations in quantum codes.

Abstract

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.