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 $r$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.12594
- arXiv:
- arXiv:2107.12594
- Bibcode:
- 2021arXiv210712594C
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Commutative Algebra;
- 94B05;
- 13P25;
- 14G50;
- 11T71
- E-Print:
- doi:10.1142/S0219498825500628