Codes from $A_m$-invariant polynomials

TL;DR

The paper develops a new class of linear codes from $A_m$-invariant polynomials over $\mathbb{F}_q[x_1,\dots,x_m]$ by leveraging Galois theory and Weil-type bounds to control zeros of message polynomials. The construction evaluates carefully chosen $A_m$-invariant polynomials on orbit representatives, yielding codes with length $n=2{q \choose m}$ and dimension $k=2(m+1)$, along with a guaranteed distance bound $d \ge n - \left(2{q \choose m}/(q-1) + 2{q-1 \choose m-1}\right)$ under suitable parameter regimes. The resulting codes match the asymptotic relative distance of Generalized Reed–Muller codes while offering a superior rate, and compare favorably to Datta–Johnsen codes by roughly doubling both length and dimension for fixed $q$. The authors also sketch a broader program to extend these ideas to arbitrary subgroups of the symmetric group, with potential refinements via geometric techniques to relax parameter constraints.

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.

Theorems & Definitions (25)

• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• proof
• Remark 2.7
• Remark 3.1