On the generalized Hamming weights of hyperbolic codes
Abstract
A hyperbolic code is an evaluation code that improves a Reed-Muller because the dimension increases while the minimum distance is not penalized. We give the necessary and sufficient conditions, based on the basic parameters of the Reed-Muller, to determine whether a Reed-Muller coincides with a hyperbolic code. Given a hyperbolic code, we find the largest Reed-Muller containing the hyperbolic code and the smallest Reed-Muller in the hyperbolic code. We then prove that similarly to Reed-Muller and Cartesian codes, the -th generalized Hamming weight and the -th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the -th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the -th footprint of a hyperbolic code that, sometimes, are sharp.
Cite
@article{arxiv.2107.12594,
title = {On the generalized Hamming weights of hyperbolic codes},
author = {Eduardo Camps-Moreno and Ignacio García-Marco and Hiram H. López and Irene Márquez-Corbella and Edgar Martínez-Moro and Eliseo Sarmiento},
journal= {arXiv preprint arXiv:2107.12594},
year = {2024}
}