On parameterized toric codes
APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, cilt.34, sa.3, ss.443-467, 2023 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 34 Sayı: 3
- Basım Tarihi: 2023
- Doi Numarası: 10.1007/s00200-021-00513-8
- Dergi Adı: APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH
- Sayfa Sayıları: ss.443-467
- Anahtar Kelimeler: Evaluation code, Lattice ideal, Multigraded Hilbert function, Parameterized code, Toric variety, Vanishing ideal
- Hacettepe Üniversitesi Adresli: Evet
Özet
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 C-alpha,C-YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of C-alpha,C-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.