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arXiv · 2502.14746

Coxeter codes: Extending the Reed-Muller family

Nolan J. Coble·Alexander Barg
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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.

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Nolan J. Coble, Alexander Barg. 2025-02-20. Coxeter codes: Extending the Reed-Muller family. https://doi.org/10.1007/s10623-025-01749-y

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