On the generalized Hamming weights of hyperbolic codes

Abstract

A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.