Codes from $A_m$-invariant polynomials

TL;DR

This paper introduces a new class of linear codes derived from the action of the alternating group on polynomial rings over finite fields, achieving better rates than similar existing codes while maintaining comparable asymptotic relative distance.

Contribution

It presents a novel construction of codes using group actions and Galois theory, improving the rate of generalized Reed-Muller codes with similar parameters.

Findings

Codes have the same asymptotic relative distance as generalized Reed-Muller codes.

New codes exhibit better rate performance.

Results are based on Galois theory and Weil bounds.

Abstract

Let $q$ be a prime power. This paper provides a new class of linear codes that arises from the action of the alternating group on $\mathbb F_q[x_1,\dots,x_m]$ combined with the ideas in (M. Datta and T. Johnsen, 2022). Compared with Generalized Reed-Muller codes with similar parameters, our codes have the same asymptotic relative distance but a better rate. Our results follow from combinations of Galois theoretical methods with Weil-type bounds for the number of points of hypersurfaces over finite fields.