Work

Minimum distance and decoding of Coxeter codes

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2607.10774

Authors: Barg, Alexander, Gashi, Qëndrim R., Xu, Tianyuan

Institutions: University of Richmond, University of Maryland, College Park, University of Prishtina

Topic: Coding theory and cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, University of Maryland

Awards: CIF: Small: Recoverable systems, codes with local reconstruction, and interaction models, CIF: Small: Quantum LDPC codes: structure and logical operations

OpenAlex ID: W7168259648

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