The Permutation Group of Reed-Solomon Codes Over Arbitrary Points
🤖 AI Summary
This work investigates the permutation group structure of Reed–Solomon codes evaluated on arbitrary point sets. By analyzing affine transformations induced by linear polynomials that preserve the evaluation set, and leveraging polynomial mappings over finite fields together with group action theory, the paper provides a unified characterization of these permutation groups. The main contribution is a rigorous proof that the permutation group consists precisely of those permutations induced by linear polynomials stabilizing the evaluation set. This framework naturally encompasses classical cases—such as evaluations over the entire finite field or over multiplicative subgroups—as special instances, while also offering a more streamlined and general proof approach that subsumes and simplifies existing results in the literature.
Application Category
📝 Abstract
In this work, we prove that the permutation group of a Reed-Solomon code is given by the polynomials of degree one that leave the set of evaluation points invariant. Our results provide a straightforward proof of the well-known cases of the permutation group of the Reed-Solomon code when the set of evaluation points is the whole finite field or the multiplicative group.
Problem
Research questions and friction points this paper is trying to address.
Reed-Solomon codes
permutation group
evaluation points
finite field
polynomial automorphisms
Innovation
Methods, ideas, or system contributions that make the work stand out.
Reed-Solomon codes
permutation group
evaluation points
degree-one polynomials
finite fields
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