On parameterized toric codes

Baran E., ŞAHİN M.

APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, cilt.34, sa.3, ss.443-467, 2023 (SCI-Expanded) identifier identifier

  • 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.