Rational points of lattice ideals on a toric variety and toric codes

ŞAHİN M.

Finite Fields and their Applications, cilt.90, 2023 (SCI-Expanded) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 90
  • Basım Tarihi: 2023
  • Doi Numarası: 10.1016/j.ffa.2023.102226
  • Dergi Adı: Finite Fields and their Applications
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, MathSciNet, zbMATH
  • Anahtar Kelimeler: Algebraic geometric codes, Generalised toric codes, Lattice ideals, Rational points
  • Hacettepe Üniversitesi Adresli: Evet

Özet

We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature.