On the affine permutation group of certain decreasing Cartesian codes

TL;DR

This paper investigates the affine permutation groups of decreasing Cartesian codes, including Reed-Solomon, Reed-Muller, and toric codes, focusing on cases involving subgroup structures within the Cartesian set.

Contribution

It characterizes the affine permutation groups of certain decreasing Cartesian codes, especially when the set includes subgroup copies, advancing understanding of their symmetry properties.

Findings

Affine permutation groups depend on subgroup structures within the Cartesian set

Characterization of affine permutations for specific decreasing Cartesian codes

Insights into the symmetry and automorphism groups of these codes

Abstract

A decreasing Cartesian code is defined by evaluating a monomial set closed under divisibility on a Cartesian set. Some well-known examples are the Reed-Solomon, Reed-Muller, and (some) toric codes. The affine permutations consist of the permutations of the code that depend on an affine transformation. In this work, we study the affine permutations of some decreasing Cartesian codes, including the case when the Cartesian set has copies of multiplicative or additive subgroups.