On the generalized Hamming weights of hyperbolic codes

被引：2

作者：

Camps-Moreno, Eduardo ^{[1
]}

Garcia-Marco, Ignacio ^{[2
]}

Lopez, Hiram H. ^{[1
]}

Marquez-Corbella, Irene ^{[2
]}

Martinez-Moro, Edgar ^{[3
]}

Sarmiento, Eliseo ^{[4
]}

机构：

[1] Virginia Tech, Dept Math, Blacksburg, VA 24061 USA

[2] Univ La Laguna, Inst Matemat & Aplicac IMAULL, Dept Matemat Estadist & IO, Tenerife, Spain

[3] Univ Valladolid, Inst Math, Valladolid, Spain

[4] Inst Politecn Nacl, Escuela Super Fis & Matemat, Mexico City, DF, Mexico

来源：

JOURNAL OF ALGEBRA AND ITS APPLICATIONS | 2024年 / 23卷 / 07期

关键词：

Reed-Muller codes; evaluation codes; hyperbolic codes; generalized Hamming weights; footprint;

D O I：

10.1142/S0219498825500628

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页数：18

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