On parameterized toric codes

Esma Baran; Mesut Şahin

doi:10.1007/s00200-021-00513-8

On parameterized toric codes

Esma Baran
, Mesut Şahin

Division of Geometry

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Let X be a complete simplicial toric variety over a finite field with a split torus T_X. For any matrix Q, we are interested in the subgroup Y_Q of T_X parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal I_L is I(Y_Q). We also give two direct algorithmic methods to compute the order of Y_Q, which is the length of the corresponding code Cα,YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of Cα,YQ, taking advantage of the parametric description of the subgroup Y_Q. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces H_ℓ generalizing the corresponding result given by Hansen.

Original language	English
Pages (from-to)	443-467
Number of pages	25
Journal	Applicable Algebra in Engineering, Communications and Computing
Volume	34
Issue number	3
DOIs	https://doi.org/10.1007/s00200-021-00513-8
Publication status	Published - May 2023

Keywords

Evaluation code
Lattice ideal
Multigraded Hilbert function
Parameterized code
Toric variety
Vanishing ideal

Access to Document

10.1007/s00200-021-00513-8

Cite this

@article{1a1ddd34512d4876bca7d5cd098eef82,

title = "On parameterized toric codes",

abstract = "Let X be a complete simplicial toric variety over a finite field with a split torus TX. For any matrix Q, we are interested in the subgroup YQ of TX parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal IL is I(YQ). We also give two direct algorithmic methods to compute the order of YQ, which is the length of the corresponding code Cα,YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of Cα,YQ, taking advantage of the parametric description of the subgroup YQ. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces Hℓ generalizing the corresponding result given by Hansen.",

keywords = "Evaluation code, Lattice ideal, Multigraded Hilbert function, Parameterized code, Toric variety, Vanishing ideal",

author = "Esma Baran and Mesut {\c S}ahin",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2023",

month = may,

doi = "10.1007/s00200-021-00513-8",

language = "English",

volume = "34",

pages = "443--467",

journal = "Applicable Algebra in Engineering, Communications and Computing",

issn = "0938-1279",

publisher = "Springer Science and Business Media Deutschland GmbH",

number = "3",

}

TY - JOUR

T1 - On parameterized toric codes

AU - Baran, Esma

AU - Şahin, Mesut

PY - 2023/5

Y1 - 2023/5

N2 - Let X be a complete simplicial toric variety over a finite field with a split torus TX. For any matrix Q, we are interested in the subgroup YQ of TX parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal IL is I(YQ). We also give two direct algorithmic methods to compute the order of YQ, which is the length of the corresponding code Cα,YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of Cα,YQ, taking advantage of the parametric description of the subgroup YQ. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces Hℓ generalizing the corresponding result given by Hansen.

AB - Let X be a complete simplicial toric variety over a finite field with a split torus TX. For any matrix Q, we are interested in the subgroup YQ of TX parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal IL is I(YQ). We also give two direct algorithmic methods to compute the order of YQ, which is the length of the corresponding code Cα,YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of Cα,YQ, taking advantage of the parametric description of the subgroup YQ. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces Hℓ generalizing the corresponding result given by Hansen.

KW - Evaluation code

KW - Lattice ideal

KW - Multigraded Hilbert function

KW - Parameterized code

KW - Toric variety

KW - Vanishing ideal

UR - https://www.scopus.com/pages/publications/85106427927

UR - https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=performanshacettepe&SrcAuth=WosAPI&KeyUT=WOS:000652926200001&DestLinkType=FullRecord&DestApp=WOS_CPL

U2 - 10.1007/s00200-021-00513-8

DO - 10.1007/s00200-021-00513-8

M3 - Article

AN - SCOPUS:85106427927

SN - 0938-1279

VL - 34

SP - 443

EP - 467

JO - Applicable Algebra in Engineering, Communications and Computing

JF - Applicable Algebra in Engineering, Communications and Computing

IS - 3

ER -

On parameterized toric codes

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this