Calculating the Minimum Distance of a Toric Code via Algebraic Algorithms

Baldemir F., ŞAHİN M.

Mathematics in Computer Science, vol.17, no.3-4, 2023 (ESCI) identifier

  • Publication Type: Article / Article
  • Volume: 17 Issue: 3-4
  • Publication Date: 2023
  • Doi Number: 10.1007/s11786-023-00566-7
  • Journal Name: Mathematics in Computer Science
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, INSPEC, zbMATH
  • Keywords: Hilbert function, Minimum distance, Toric code
  • Hacettepe University Affiliated: Yes


Toric codes are examples of evaluation codes. They are produced by evaluating homogeous polynomials of a fixed degree at the Fq -rational points of a subset Y of a toric variety X. These codes reveal how algebraic geometry and coding theory are interrelated. The minimum distance of a code is the minimum number of nonzero entries in the codewords of the code. Let I(Y) be the ideal generated by all homogeneous polynomials vanishing at all the points of Y, which is also known as the vanishing ideal of Y. We give three algebraic algorithms computing the minimum distance by using commutative algebraic tools such as the multigraded Hilbert polynomials of ideals obtained from I(Y) and zero divisors f of I(Y), and primary decomposition of I(Y), for finding a homogeneous polynomial f among all homogeneous polynomials of the same degree which has the maximum number of roots on Y.