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arXiv · 2607.10774

Minimum distance and decoding of Coxeter codes

Alexander Barg·Qëndrim R. Gashi·Tianyuan Xu
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Abstract

A binary Coxeter code associated with a finite Coxeter system $(W,S)$ is an ${\mathbb F}_2$-linear span of indicators of standard cosets of a fixed rank. Coxeter codes, introduced in a recent paper by N. Coble and A. Barg, are a generalization of Reed--Muller codes which arise when $W={\mathbb Z}_2^m$ is the Coxeter group of type $mA_1$. In that paper, the authors proposed a conjectural value for the minimum distance of a general Coxeter code. This conjecture is proved in the present work. As a consequence, we obtain a Coxeter-theoretic generalization of Reed's majority-logic decoding algorithm for Reed--Muller codes.

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Alexander Barg, Qëndrim R. Gashi, Tianyuan Xu. 2026-07-12. Minimum distance and decoding of Coxeter codes. https://arxiv.org/abs/2607.10774

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