On the generalized Hamming weights of hyperbolic codes

TL;DR

This paper investigates the properties of hyperbolic codes, especially their generalized Hamming weights, establishing conditions for their relation to Reed-Muller codes and providing bounds for their footprints.

Contribution

It characterizes when Reed-Muller codes are hyperbolic codes, finds extremal Reed-Muller codes within hyperbolic codes, and provides bounds for hyperbolic code footprints.

Findings

Hyperbolic codes improve Reed-Muller codes without reducing minimum distance.

The $r$-th generalized Hamming weight of hyperbolic codes matches that of Reed-Muller codes.

Upper and lower bounds for the $r$-th footprint of hyperbolic codes are established.

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 $r$-th generalized Hamming weight and the $r$-th footprint of the hyperbolic code coincide. Unlike Reed-Muller and Cartesian, determining the $r$-th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the $r$-th footprint of a hyperbolic code that, sometimes, are sharp.