Codes from \(A_m\)-invariant polynomials

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 \({\mathbb {F}}_q[x_1,\dots ,x_m]\) ${\mathbb{F}}_q[x_1,\cdots ,x_m]$ combined with the ideas in Datta and Johnsen (Des Codes Cryptogr 91(3):747–761, 2023). Compared with Generalized Reed–Muller codes with analogous 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 hypersurfaces.