On parameterized toric codes

Baran E., ŞAHİN M.

APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING, vol.34, no.3, pp.443-467, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 34 Issue: 3
  • Publication Date: 2023
  • Doi Number: 10.1007/s00200-021-00513-8
  • Journal Name: APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH
  • Page Numbers: pp.443-467
  • Keywords: Evaluation code, Lattice ideal, Multigraded Hilbert function, Parameterized code, Toric variety, Vanishing ideal
  • Hacettepe University Affiliated: Yes

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 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.