Reed-Muller type codes over a combinatorial simplex: an algebraic description

2026-06-01

AI Summary

This work investigates CAP codes—a class of high-rate Reed–Muller-type codes defined on combinatorial simplices—from the perspective of commutative algebra. By establishing, for the first time, a universal Gröbner basis for the vanishing ideal of a combinatorial simplex, the study connects its footprint to the generalized Hamming weights of CAP codes and derives an explicit formula for their minimum distance. Furthermore, a generating set for the dual code is constructed, and it is shown that under certain conditions, the permutation group of the dual code is the full symmetric group. These results systematically elucidate the algebraic structure, distance properties, and symmetry of CAP codes, offering new tools for the theoretical analysis of high-dimensional polynomial evaluation codes.

Abstract

Given an ordered set $B$ of a finite field, a combinatorial simplex over $B$ is defined as the set of vectors such that the positions of the entries, with respect to $B$, sum up to a fixed integer. CAP codes are Reed-Muller type codes defined over a combinatorial simplex. They were recently introduced by Kopparty et al. as a high-rate alternative to classical Reed-Muller codes, capable of achieving arbitrarily high rates close to one for any fixed minimum distance. In this paper, we use tools from commutative algebra to analyze a combinatorial simplex and its associated CAP code. We give a universal Gröbner basis for the vanishing ideal of a combinatorial simplex. We describe the generalized Hamming weights of a CAP code in terms of the footprint of the vanishing ideal. For the minimum distance case, we proved a closed formula. We give a set of polynomials whose evaluations on the combinatorial simplex generate the dual of the CAP code. We describe the affine permutations that leave invariant a combinatorial simplex and use this information to prove that, in some cases, the permutation group of a CAP code is a symmetric group.

Problem

• Reed-Muller codes
• combinatorial simplex
• CAP codes
• vanishing ideal
• generalized Hamming weights

Innovation

• combinatorial simplex
• CAP codes
• vanishing ideal
• Gröbner basis
• generalized Hamming weights