On the generalized Hamming weights of hyperbolic codes

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 C, we find the largest Reed-Muller code contained in C and the smallest Reed-Muller code containing C. We then prove that similar to Reed-Muller and affine Cartesian codes, the rth generalized Hamming weight and the rth footprint of the hyperbolic code coincide. Unlike for Reed-Muller and affine Cartesian codes, determining the rth footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the rth footprint of a hyperbolic code that, sometimes, are sharp.