Coxeter codes: Extending the Reed-Muller family

• URL: http://arxiv.org/abs/2502.14746v2
• Date: Wed, 11 Jun 2025 03:40:56 GMT
• Title: Coxeter codes: Extending the Reed-Muller family
• Authors: Nolan J. Coble, Alexander Barg,
• Abstract summary: We introduce a class of binary linear codes that generalizes the RM family by replacing the domain $mathbbZm$ with an arbitrary finite Coxeter group.<n> Coxeter codes also give rise to a family of quantum codes for which closed diagonal $Z$ rotations can perform non-trivial logic.
• Score: 59.90381090395222
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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