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 F_q [x_1,…, 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.
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