On Reed-Muller-Type Codes Associated to Proto-Exact Categories
Abstract
In this paper, we show how to associate a family of binary linear block codes to the class of admissibly cofinitary objects in any proto-exact category with rank function. This family naturally generalizes the well-studied family of binary Reed-Muller codes, and recovers this classical family for suitable choices of category. We also expand the list of known examples of proto-abelian categories to include incidence categories of posets. As an application of this result, we construct a binormal, combinatorial, proto-abelian category of Kashiwara crystals associated to any root system. In the case that the proto-exact category is a category of 𝔽1-representations of quivers, we show how to reconstruct our family via the language of algebraic geometry codes, in particular motivating an extension to arbitrary finite fields. We provide explicit computations of the dimension and minimum distance for select families of 𝔽1-representations of quivers.
Related
- Common zeros of homogeneous polynomials and the weight hierarchy of projective Reed-Muller codes
- The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes
- Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs
- Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$
- Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions
BibTeX
@misc{edison2026reedmullertype,
title = {{On Reed-Muller-Type Codes Associated to Proto-Exact Categories}},
author = {Jeremy Edison and Alexander Sistko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33903},
eprint = {2610.33903},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33903},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Reed-Muller-Type Codes Associated to Proto-Exact Categories
AU - Jeremy Edison
AU - Alexander Sistko
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33903
UR - https://arxiv.org/abs/2610.33903
AB - In this paper, we show how to associate a family of binary linear block codes to the class of admissibly cofinitary objects in any proto-exact category with rank function. This family naturally generalizes the well-studied family of binary Reed-Muller codes, and recovers this classical family for suitable choices of category. We also expand the list of known examples of proto-abelian categories to include incidence categories of posets. As an application of this result, we construct a binormal, combinatorial, proto-abelian category of Kashiwara crystals associated to any root system. In the case that the proto-exact category is a category of 𝔽_1-representations of quivers, we show how to reconstruct our family via the language of algebraic geometry codes, in particular motivating an extension to arbitrary finite fields. We provide explicit computations of the dimension and minimum distance for select families of 𝔽_1-representations of quivers.
ER -