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 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.
Communicated by E. Gorla
Dedicated to Joachim Rosenthal on the occasion of his 60th birthday. We thank Prof. Rosenthal for his selfless, continuous, and endless support to shape the coding theory and cryptography community.